NTCIR12-MathWiki-1rate: 0

$i=0\ldots n-1$

NTCIR12-MathWiki-1rate: 0

$\begin{aligned}\displaystyle{\left({{-n}\atop{k}}\right)}&\displaystyle=\frac{% -n\cdot-(n+1)\dots-(n+k-2)\cdot-(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}\;\frac{n\cdot(n+1)\cdot(n+2)\cdots(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}{\left({{n+k-1}\atop{k}}\right)}\\ &\displaystyle=(-1)^{k}\left(\!\!{\left({{n}\atop{k}}\right)}\!\!\right)\;.% \end{aligned}$

NTCIR12-MathWiki-1rate: 1

$j_{2A}\Big(\tfrac{\sqrt{-58}}{2}\Big)=396^{4},\qquad\quad e^{\pi\sqrt{58}}% \approx 396^{4}-104.00000017\dots$

NTCIR12-MathWiki-1rate: 0

$i=0\ldots n-2.$

NTCIR12-MathWiki-1rate: 0

$\begin{matrix}\,\text{Total}\\ \,\text{resources}\\ \,\text{available}\end{matrix}\ \ \ =\ \ \ \begin{matrix}\,\text{Resources}\\ \,\text{at start}\\ \,\text{of innings}\end{matrix}\ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \ \begin{% matrix}\,\text{Resources lost by}\\ \,\text{first interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ % \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{second interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ % \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{third interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ % \ \,\text{etc}...$

NTCIR12-MathWiki-1rate: 0

$\{0,1\dots p-1\}$

NTCIR12-MathWiki-1rate: 0

$\displaystyle 2x-2x-y$

NTCIR12-MathWiki-1rate: 3

$\psi=\frac{1-\sqrt{5}}{2}=1-\varphi=-{1\over\varphi}\approx-0.61803\,39887\cdots$

NTCIR12-MathWiki-1rate: 0

$\;\epsilon_{\mathrm{n}}=\epsilon-K_{\mathrm{p}}...........(16)$

NTCIR12-MathWiki-1rate: 0

$2π-2φ$

NTCIR12-MathWiki-1rate: 2

$\tau_{5}=\frac{\beta-\alpha}{60^{\circ}}\approx-0.01667\alpha+0.01667\beta$

NTCIR12-MathWiki-1rate: 0

$4t-2n$

NTCIR12-MathWiki-1rate: 2

$\varphi=\frac{1-\sqrt{5}}{2}=-0.6180\,339887\dots$

NTCIR12-MathWiki-1rate: 1

$\begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx(5x)^{3}-6.0% 00\,010\dots\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx(5x)^{3}-6.000\,000\,010% \dots\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,0% 61\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,% 000\,000\,034\dots\end{aligned}$

NTCIR12-MathWiki-1rate: 0

$H(X_{1},...,X_{n})=-\sum_{x_{1}}...\sum_{x_{n}}P(x_{1},...,x_{n})\log_{2}[P(x_% {1},...,x_{n})]\!$

NTCIR12-MathWiki-1rate: 2

$M=-6+12/\sqrt{2}-8/\sqrt{3}+6/2-24/\sqrt{5}+\cdots=-1.74756\dots.$

NTCIR12-MathWiki-1rate: 0

$\mathbf{m}=-\gamma\mathbf{L}$

NTCIR12-MathWiki-1rate: 2

$(\pi+20)^{i}=-0.9999999992\ldots-i\cdot 0.000039\ldots\approx-1$

NTCIR12-MathWiki-1rate: 2

$\scriptstyle\zeta^{\prime}(2)\,\,\text{= Derivative of }\zeta(2)=-\sum\limits_% {n=2}^{\infty}\frac{\ln n}{n^{2}}=-0.9375482543\ldots$

NTCIR12-MathWiki-1rate: 0

$\{0,i,i-1,-i,i-1,-i...\}\,$

NTCIR12-MathWiki-1rate: 2

$\sqrt[5]{-2}\,=-1.148698354\ldots$

NTCIR12-MathWiki-1rate: 2

$\zeta^{\prime}(-1)=\frac{1}{12}-\ln A\approx-0.1654211437\ldots$

NTCIR12-MathWiki-1rate: 1

$1-1\sqrt{2}=-0.41421\ldots$

NTCIR12-MathWiki-1rate: 2

$41-29\sqrt{2}=-0.01219\ldots$

NTCIR12-MathWiki-1rate: 4

$1393-985\sqrt{2}=-0.00035\ldots$

NTCIR12-MathWiki-1rate: 0

$2n-h-2$

NTCIR12-MathWiki-1rate: 1

$c_{2}=-\frac{1}{6}\quad;\quad c_{3}=-\frac{1}{108}\quad;\quad c_{4}=\frac{7}{3% 240}\quad;\quad c_{5}=-\frac{19}{48600}\ \dots$

NTCIR12-MathWiki-1rate: 4

$-0.026838601\ldots$

NTCIR12-MathWiki-1rate: 0

$2-2g-n$

NTCIR12-MathWiki-1rate: 2

$x=-1.84208\dots$

NTCIR12-MathWiki-1rate: 0

$1,1,2,3,1,-2,-3,-5,-2,-3,\ldots\,\text{ for the signs }+,+,+,-,-,+,-,-,\ldots.$

NTCIR12-MathWiki-1rate: 2

$\theta^{\prime}(0)=-\frac{\ln\pi+\gamma+\pi/2+3\ln 2}{2}=-2.6860917\ldots$

NTCIR12-MathWiki-1rate: 2

$\pm 17.8455995405\ldots$

NTCIR12-MathWiki-1rate: 0

$e^{-\delta\tau}$

NTCIR12-MathWiki-1rate: 0

$2a-R$

NTCIR12-MathWiki-1rate: 2

$\lim_{k\to\infty}c_{2}(k)=-2.157782996659\ldots.$

NTCIR12-MathWiki-1rate: 2

$e^{\pi\sqrt{58}}=396^{4}-104.000000177\dots.$

NTCIR12-MathWiki-1rate: 0

$2-2g-n$

NTCIR12-MathWiki-1rate: 2

$\gamma_{1}\!\left(\!\frac{1}{\,2\,}\!\right)=-2\gamma\ln 2-\ln^{2}\!2+\gamma_{% 1}\,=\,-1.353459680\ldots$

NTCIR12-MathWiki-1rate: 0

$-\Lambda\hbar$

NTCIR12-MathWiki-1rate: 0

$-\omega\delta$

NTCIR12-MathWiki-1rate: 2

$z_{1}=-0.4121345\ldots+i0.5978119\ldots$

NTCIR12-MathWiki-1rate: 2

$z_{2}=-1.4455692\ldots+i0.6992608\ldots$

NTCIR12-MathWiki-1rate: 0

$2\pi-\nu$

NTCIR12-MathWiki-1rate: 0

$\phi_{1}=-30^{\circ}...+30^{\circ}$

NTCIR12-MathWiki-1rate: 0

$\frac{1-e^{-\lambda\theta}}{1-e^{-2\pi\lambda}}$

NTCIR12-MathWiki-10rate: 0

$\sum_{k=0}^{\infty}a_{k}=\sum_{k=0}^{\infty}(-1)^{k}(k+1),$

NTCIR12-MathWiki-10rate: 1

$\zeta(s,\alpha)=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^{s}}=\frac{\alpha^{1-s}% }{s-1}+\frac{1}{2\alpha^{s}}+2\int_{0}^{\infty}\frac{\sin\left(s\arctan\frac{t% }{\alpha}\right)}{(\alpha^{2}+t^{2})^{\frac{s}{2}}}\frac{dt}{e^{2\pi t}-1}.$

NTCIR12-MathWiki-10rate: 0

$f(z)=\sum_{n=0}^{\infty}(z-a)^{n}{1\over 2\pi i}\int_{C}{f(w)\over(w-a)^{n+1}}% \,\mathrm{d}w.$

NTCIR12-MathWiki-10rate: 1

$S_{\nu}(x)=\sum_{k=0}^{\infty}\frac{\sin((2k+1)\pi x)}{(2k+1)^{\nu}}$

NTCIR12-MathWiki-10rate: 1

${\rm blanc}(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}},$

NTCIR12-MathWiki-10rate: 1

$L(s)\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}\qquad\qquad L(1-s)=L(% s)\Gamma(s)2^{s}\pi^{-s}\sin\frac{\pi s}{2},$

NTCIR12-MathWiki-10rate: 2

$\Phi(z,s,\alpha)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+\alpha)^{s}}.$

NTCIR12-MathWiki-10rate: 1

$M(a,b,z)=\sum_{n=0}^{\infty}\frac{a^{(n)}z^{n}}{b^{(n)}n!}={}_{1}F_{1}(a;b;z),$

NTCIR12-MathWiki-10rate: 1

$Z(\lambda,\nu)=\sum_{j=0}^{\infty}\frac{\lambda^{j}}{(j!)^{\nu}}.$

NTCIR12-MathWiki-10rate: 0

$\sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\frac{p(n)}{q(n)},$

NTCIR12-MathWiki-10rate: 1

$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}},$

NTCIR12-MathWiki-10rate: 1

$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}$

NTCIR12-MathWiki-10rate: 1

$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}.$

NTCIR12-MathWiki-10rate: 1

$\frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.$

NTCIR12-MathWiki-10rate: 1

$F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}$

NTCIR12-MathWiki-10rate: 1

$f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\frac{f(x_{j})}{q^{\prime}(x_{j})}.$

NTCIR12-MathWiki-10rate: 1

$\textstyle{f(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}}}$

NTCIR12-MathWiki-10rate: 1

$\frac{1}{n}=\sum_{r=1}^{\infty}\frac{1}{(n+1)^{r}}.$

NTCIR12-MathWiki-10rate: 1

$\sum_{n=0}^{\infty}\frac{1}{(n+a)}$

NTCIR12-MathWiki-10rate: 0

$\,{}_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)=\sum_{n=0}^{\infty}% \frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}\,\frac{z^{n}}{% n!}$

NTCIR12-MathWiki-10rate: 1

$L(\chi,s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}$

NTCIR12-MathWiki-10rate: 0

${}_{p}F_{q}(a_{1},\dots,a_{p};c_{1},\dots,c_{q};z)=\sum_{n=0}^{\infty}\frac{(a% _{1})_{n}\cdots(a_{p})_{n}}{(c_{1})_{n}\cdots(c_{q})_{n}}\frac{z^{n}}{n!}$

NTCIR12-MathWiki-10rate: 1

$x=\sum_{n=0}^{\infty}\frac{a_{n}}{10^{n}}.$

NTCIR12-MathWiki-10rate: 1

$\zeta(s,q)=\sum_{n=0}^{\infty}\frac{1}{(q+n)^{s}}.$

NTCIR12-MathWiki-10rate: 2

$\Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}$

NTCIR12-MathWiki-10rate: 1

$\textstyle\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$

NTCIR12-MathWiki-10rate: 1

${}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^% {n}}{n!}.$

NTCIR12-MathWiki-10rate: 4

$L(\lambda,\alpha,s)=\sum_{n=0}^{\infty}\frac{\exp(2\pi i\lambda n)}{(n+\alpha)% ^{s}}.$

NTCIR12-MathWiki-10rate: 1

$\frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.$

NTCIR12-MathWiki-10rate: 1

$\displaystyle f(x)=\sum_{k=1}^{\infty}\frac{\sin(2^{k}x)}{\sqrt{2}^{k}}$

NTCIR12-MathWiki-10rate: 1

$Z(P,Q,s)=\sum_{n=1}^{\infty}\frac{f_{n}(P)f_{n}(Q)}{\lambda_{n}^{s}}$

NTCIR12-MathWiki-10rate: 1

$L(E,s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.$

NTCIR12-MathWiki-10rate: 1

$L(s,E)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.$

NTCIR12-MathWiki-10rate: 2

$\zeta(s,t)=\sum_{n=1}^{\infty}\frac{H_{n,t}}{(n+1)^{s}}$

NTCIR12-MathWiki-10rate: 0

$\exp(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}.$

NTCIR12-MathWiki-10rate: 1

$\sum_{n=0}^{\infty}\frac{P(n)}{\alpha^{n}}=\frac{\alpha^{2}(\alpha+1)}{\alpha^% {3}-\alpha-1}.$

NTCIR12-MathWiki-10rate: 1

$g(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}$

NTCIR12-MathWiki-10rate: 1

$\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)^{m+1}}=\frac{1}{(-2)^{m+1}m!}\left[% \psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right]$

NTCIR12-MathWiki-10rate: 1

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},$

NTCIR12-MathWiki-10rate: 1

$e_{q}^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{[n]_{q}!}.$

NTCIR12-MathWiki-10rate: 1

$A(w)e_{q}(zw)=\sum_{n=0}^{\infty}\frac{p_{n}(z)}{[n]_{q}!}w^{n}$

NTCIR12-MathWiki-10rate: 1

$\zeta(s,a)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}\!$

NTCIR12-MathWiki-10rate: 2

$\Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}\!$

NTCIR12-MathWiki-10rate: 1

$L(s,\Delta)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$

NTCIR12-MathWiki-10rate: 1

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$

NTCIR12-MathWiki-10rate: 1

$\cos(\varphi)=\sum_{n=0}^{\infty}\frac{(-\varphi^{2})^{n}}{(2n)!},$

NTCIR12-MathWiki-10rate: 1

$G(x,z;\lambda)=\sum_{i=1}^{n}\frac{e_{i}(x)f_{i}^{*}(z)}{\lambda-\lambda_{i}}.$

NTCIR12-MathWiki-10rate: 1

$\widehat{\theta}_{F}(z)=\sum_{k=0}^{\infty}R_{F}(k)\exp(2\pi ikz),$

NTCIR12-MathWiki-10rate: 1

$\psi_{1}(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^{2}},$

NTCIR12-MathWiki-10rate: 1

$F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}$

NTCIR12-MathWiki-11rate: 4

$\textstyle{ax^{2}+bx+c=0,a\neq 0}$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0,$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0\,$

NTCIR12-MathWiki-11rate: 3

$ax^{2}+bx=c$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0,\,$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0,\,\text{ with }a\neq 0,$

NTCIR12-MathWiki-11rate: 4

$ax^{3}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$as^{2}+bs+c=0\,$

NTCIR12-MathWiki-11rate: 4

$ad^{2}+bd+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$\ ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0\ .$

NTCIR12-MathWiki-11rate: 2

$ax+by+c=0\,$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0\,$

NTCIR12-MathWiki-11rate: 4

$\ Ax^{2}+Bx+C=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0,\,\!$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0.$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$an^{2}+bn+q=0$

NTCIR12-MathWiki-11rate: 2

$ax+by+c=0\,$

NTCIR12-MathWiki-11rate: 4

$a\lambda^{2}+b\lambda+c=0\;$

NTCIR12-MathWiki-11rate: 2

$ax+by+c=0\,$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0,\,\!$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-11rate: 4

$ax^{2}+bx+c=0$

NTCIR12-MathWiki-12rate: 2

$O(\log n)$

NTCIR12-MathWiki-12rate: 1

$O(mn)$

NTCIR12-MathWiki-12rate: 1

$O(mn)$

NTCIR12-MathWiki-12rate: 0

$\Theta(mnp)$

NTCIR12-MathWiki-12rate: 3

$O(\eta*\log{n})$

NTCIR12-MathWiki-12rate: 1

$O(mn)$

NTCIR12-MathWiki-12rate: 4

$O(VE\log V)$

NTCIR12-MathWiki-12rate: 2

$O(\log n)$

NTCIR12-MathWiki-12rate: 2

$O(\log n)$

NTCIR12-MathWiki-12rate: 0

$O(lgn)$

NTCIR12-MathWiki-12rate: 0

$O(lgn)$

NTCIR12-MathWiki-12rate: 3

$O(n\log k)$

NTCIR12-MathWiki-12rate: 3

$O(nk\log k)$

NTCIR12-MathWiki-12rate: 2

$Q_{D}\left(n\right)=O(Q_{S}\left(n\right)\log\left(n\right))\,\!$

NTCIR12-MathWiki-12rate: 3

$O(E\log V)$

NTCIR12-MathWiki-12rate: 1

$O(mr)\,$

NTCIR12-MathWiki-12rate: 4

$s_{f_{2}}(n)=O\left(M(n)\log n\right).$

NTCIR12-MathWiki-12rate: 3

$O\left(M(m)\log^{2}m\right)=O\left(M(n)\log n\right).\,$

NTCIR12-MathWiki-12rate: 1

$O(MST)$

NTCIR12-MathWiki-12rate: 3

$O(n\log h)$

NTCIR12-MathWiki-12rate: 1

$O(KNM)$

NTCIR12-MathWiki-12rate: 4

$O(KN\log N)$

NTCIR12-MathWiki-12rate: 1

$O\left(m^{g(k)}n^{k}\right)$

NTCIR12-MathWiki-12rate: 3

$O(d\log{n})$

NTCIR12-MathWiki-12rate: 3

$O(nk\,\log(n))$

NTCIR12-MathWiki-12rate: 4

$O(mn\log m)$

NTCIR12-MathWiki-12rate: 0

$\tilde{O}(m)$

NTCIR12-MathWiki-12rate: 3

$O(mn\log p)=O(n\log n)$

NTCIR12-MathWiki-12rate: 3

$O(k\log n)$

NTCIR12-MathWiki-12rate: 2

$O(Tm)=O(n^{2}m\log n)$

NTCIR12-MathWiki-12rate: 3

$O(m\log n\log\log n)$

NTCIR12-MathWiki-12rate: 3

$O(\lambda_{\delta}(n)\log n)$

NTCIR12-MathWiki-12rate: 3

$O(n\log m)$

NTCIR12-MathWiki-12rate: 3

$O(n\log m)$

NTCIR12-MathWiki-12rate: 3

$O(m\log n)$

NTCIR12-MathWiki-12rate: 1

$O(m+\log n)$

NTCIR12-MathWiki-12rate: 1

$O(d^{5}n\log^{3}B)\,$

NTCIR12-MathWiki-12rate: 0

$O(mnp)$

NTCIR12-MathWiki-12rate: 2

$O(\log^{2}n\log\log n)$

NTCIR12-MathWiki-12rate: 1

$O(VE\log V\log(VC))$

NTCIR12-MathWiki-12rate: 3

$O(n\log n)$

NTCIR12-MathWiki-12rate: 1

$O(mn)$

NTCIR12-MathWiki-12rate: 0

$O(Ns\kappa)$

NTCIR12-MathWiki-12rate: 3

$O(mn)=O(n^{3}\log n)$

NTCIR12-MathWiki-12rate: 0

$O(ckN)$

NTCIR12-MathWiki-12rate: 3

$O(n\log k)$

NTCIR12-MathWiki-12rate: 2

$O\left(mnr^{2}\log\frac{1}{\epsilon}\right)$

NTCIR12-MathWiki-12rate: 2

$O(pn(m+n\log n))$

NTCIR12-MathWiki-12rate: 2

$O(\log n)$

NTCIR12-MathWiki-12rate: 2

$O(\log n)$

NTCIR12-MathWiki-12rate: 2

$O(\log{n})$

NTCIR12-MathWiki-12rate: 2

$O(\log{n})$

NTCIR12-MathWiki-12rate: 2

$T_{\mathrm{amortized}}(m)=O(m\log n)$

NTCIR12-MathWiki-12rate: 0

$O(m)$

NTCIR12-MathWiki-12rate: 0

$O(m)$

NTCIR12-MathWiki-12rate: 0

$O(m)$

NTCIR12-MathWiki-12rate: 0

$O(m)$

NTCIR12-MathWiki-12rate: 1

$O(mn)$

NTCIR12-MathWiki-12rate: 2

$O(m\sqrt{n}\log n)$

NTCIR12-MathWiki-13rate: 0

$I_{n}^{f}=\left(A_{n}^{f}\right)^{2},$

NTCIR12-MathWiki-13rate: 0

$A=(r^{n}-1)/B=3$

NTCIR12-MathWiki-13rate: 0

$R=(A+I)^{n}$

NTCIR12-MathWiki-13rate: 0

$M_{1}=M\oplus P=(M\setminus P)\cup(P\setminus M)$

NTCIR12-MathWiki-13rate: 1

$x\oplus y=(x\vee y)\wedge\neg{(x\wedge y)}$

NTCIR12-MathWiki-13rate: 0

$P^{\prime}=P\oplus\Delta$

NTCIR12-MathWiki-13rate: 0

$P^{\prime}(A,B)=A\oplus B$

NTCIR12-MathWiki-13rate: 0

$(A\times B)^{c}=(A^{c}\times B^{c})\cup(A^{c}\times B)\cup(A\times B^{c}).$

NTCIR12-MathWiki-13rate: 0

$A\bullet B=(A\oplus B)\ominus B,\,$

NTCIR12-MathWiki-13rate: 0

$\mathit{B}=(x^{b_{i}})$

NTCIR12-MathWiki-13rate: 0

$x\oplus y=(x+y)/2$

NTCIR12-MathWiki-13rate: 0

$\left(A\cup B\right)^{c}=A^{c}\cap B^{c}.$

NTCIR12-MathWiki-13rate: 0

$\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}}$

NTCIR12-MathWiki-13rate: 0

$Y^{\prime}=Y\oplus N$

NTCIR12-MathWiki-13rate: 0

$A\cap B=\overline{\overline{A}\cup\overline{B}}$

NTCIR12-MathWiki-13rate: 0

$M=T\oplus P$

NTCIR12-MathWiki-13rate: 1

$A^{c}\cup B^{c}\subseteq(A\cap B)^{c}$

NTCIR12-MathWiki-13rate: 1

$A^{c}\cup B^{c}\subseteq(A\cap B)^{c}$

NTCIR12-MathWiki-13rate: 1

$A^{c}\cup B^{c}\subseteq(A\cap B)^{c}$

NTCIR12-MathWiki-13rate: 0

$(A\cup B)^{c}=A^{c}\cap B^{c}$

NTCIR12-MathWiki-13rate: 1

$A\oplus B=\{z\in E\mid(B^{s})_{z}\cap A\neq\varnothing\}$

NTCIR12-MathWiki-13rate: 0

$Q=\left(A^{T}A\right)^{-1}$

NTCIR12-MathWiki-13rate: 0

$B^{c}\subseteq A^{c}$

NTCIR12-MathWiki-13rate: 0

$A\circ B=(AB+BA)/2$

NTCIR12-MathWiki-13rate: 0

$K^{\ominus}=\frac{[R]^{\rho}[S]^{\sigma}...}{[A]^{\alpha}[B]^{\beta}...}\times% \frac{{\gamma_{R}}^{\rho}{\gamma_{S}}^{\sigma}...}{{\gamma_{A}}^{\alpha}{% \gamma_{B}}^{\beta}...}\times\frac{\left({c^{\ominus}_{A}}\right)^{\alpha}% \left({c^{\ominus}_{B}}\right)^{\beta}...}{\left({c^{\ominus}_{R}}\right)^{% \rho}\left({c^{\ominus}_{S}}\right)^{\sigma}...}=Q^{E}\Gamma C^{0}$

NTCIR12-MathWiki-13rate: 0

$(A\ominus B)\ominus C=A\ominus(B\oplus C)$

NTCIR12-MathWiki-13rate: 0

$Q=\left(A^{T}A\right)^{-1}$

NTCIR12-MathWiki-13rate: 0

$(A\oplus B)$

NTCIR12-MathWiki-13rate: 0

$\ A\vee B:=(A\rightarrow B)\rightarrow B.$

NTCIR12-MathWiki-13rate: 0

$I^{p,\bullet}=J\oplus K$

NTCIR12-MathWiki-13rate: 1

$\mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))$

NTCIR12-MathWiki-13rate: 0

$A\odot B=(A\ominus C)\cap(A^{c}\ominus D)$

NTCIR12-MathWiki-13rate: 0

$x=(A^{T}A)^{-1}A^{T}b.$

NTCIR12-MathWiki-13rate: 1

$A\triangle B=(A\cup B)-(A\cap B)$

NTCIR12-MathWiki-13rate: 0

$U=(A^{T}A)^{1/2}A^{-1}$

NTCIR12-MathWiki-13rate: 0

$\alpha\ominus\beta:=\alpha\oplus(\ominus\beta)$

NTCIR12-MathWiki-13rate: 0

$\epsilon=(\log\log n)^{c}$

NTCIR12-MathWiki-13rate: 0

$A\equiv B\quad=\quad(A\multimap B)\&(B\multimap A)$

NTCIR12-MathWiki-13rate: 0

$w=(A^{T}A)^{-1}A^{T}b.$

NTCIR12-MathWiki-13rate: 0

$B\to(A\lor B)$

NTCIR12-MathWiki-13rate: 0

$B\to(A\lor B)$

NTCIR12-MathWiki-13rate: 0

$\overline{A\oplus B}$

NTCIR12-MathWiki-13rate: 1

$\,\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))$

NTCIR12-MathWiki-13rate: 1

$\mathrm{v}=(A^{T}A)^{-1}A^{T}b$

NTCIR12-MathWiki-13rate: 2

$A\oplus B=(A^{c}\ominus B^{s})^{c}$

NTCIR12-MathWiki-13rate: 1

$W=(A^{T}A)^{-1}A^{T}$

NTCIR12-MathWiki-13rate: 2

$A\bullet B=(A^{c}\circ B^{s})^{c}$

NTCIR12-MathWiki-13rate: 0

$V=U\oplus W$

NTCIR12-MathWiki-13rate: 0

$A=R\oplus V,$

NTCIR12-MathWiki-13rate: 0

$A\,\triangle\,B=(A\setminus B)\cup(B\setminus A)$

NTCIR12-MathWiki-13rate: 0

$A\cup B=(A\,\triangle\,B)\,\triangle\,(A\cap B)$

NTCIR12-MathWiki-13rate: 0

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$

NTCIR12-MathWiki-13rate: 0

$A\oplus B=\mathcal{N}.$

NTCIR12-MathWiki-13rate: 0

$A\,\Delta\,B=(A\setminus B)\cup(B\setminus A).$

NTCIR12-MathWiki-13rate: 2

$\pi\oplus\sigma=\operatorname{rev}(\operatorname{rev}(\sigma)\ominus% \operatorname{rev}(\pi))$

NTCIR12-MathWiki-13rate: 0

$D=y\oplus x$

NTCIR12-MathWiki-13rate: 0

$A\,\cup\,B=(A\,\triangle\,B)\,\triangle\,(A\cap B)$

NTCIR12-MathWiki-13rate: 0

$A\oplus B.$

NTCIR12-MathWiki-13rate: 0

$R\oplus G=(\,\text{dom }G\ntriangleleft R)\cup G$

NTCIR12-MathWiki-13rate: 0

$C=A\oplus B$

NTCIR12-MathWiki-13rate: 0

$A\cdot B=\overline{\overline{A}+\overline{B}}$

NTCIR12-MathWiki-13rate: 1

$A\cup B=\left(A^{C}\cap B^{C}\right)^{C}$

NTCIR12-MathWiki-13rate: 0

$(A\oplus B)\oplus A=A\oplus(A\oplus B)$

NTCIR12-MathWiki-13rate: 0

$A\oplus B=B\oplus A$

NTCIR12-MathWiki-14rate: 1

$symbol\omega=(\dot{\alpha}\sin\beta\sin\gamma+\dot{\beta}\cos\gamma)I+(\dot{% \alpha}\sin\beta\cos\gamma-\dot{\beta}\sin\gamma)J+(\dot{\alpha}\cos\beta+\dot% {\gamma})K$

NTCIR12-MathWiki-14rate: 0

$F_{total}=\mu mg\cos{\theta}+mg\sin{\theta}$

NTCIR12-MathWiki-14rate: 0

$\cos C=-\cos A$

NTCIR12-MathWiki-14rate: 0

$abc\sqrt{1-\cos^{2}\alpha-\cos^{2}\beta-\cos^{2}\gamma+2\cos\alpha\cos\beta% \cos\gamma}$

NTCIR12-MathWiki-14rate: 1

$y(u,v)=-\cos\theta\,\sinh v\,\cos u+\sin\theta\,\cosh v\,\sin u$

NTCIR12-MathWiki-14rate: 0

$\tan\lambda={\sin\alpha\cos\varepsilon+\tan\delta\sin\varepsilon\over\cos% \alpha};\qquad\qquad\begin{cases}\cos\beta\sin\lambda=\cos\delta\sin\alpha\cos% \varepsilon+\sin\delta\sin\varepsilon;\\ \cos\beta\cos\lambda=\cos\delta\cos\alpha.\end{cases}$

NTCIR12-MathWiki-14rate: 1

$\sin\delta=\sin\beta\cos\varepsilon+\cos\beta\sin\varepsilon\sin\lambda$

NTCIR12-MathWiki-14rate: 1

$\sin a=\sin\phi_{o}\sin\delta+\cos\phi_{o}\cos\delta\cos h$

NTCIR12-MathWiki-14rate: 0

$dG_{T,P}=-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}+W^{\prime}.\,$

NTCIR12-MathWiki-14rate: 0

$\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\,$

NTCIR12-MathWiki-14rate: 0

$\cos(x-y)=\cos x\cos y-\sin x\sin y$

NTCIR12-MathWiki-14rate: 0

$\begin{bmatrix}\cos\theta\cos\psi&-\cos\phi\sin\psi+\sin\phi\sin\theta\cos\psi% &\sin\phi\sin\psi+\cos\phi\sin\theta\cos\psi\\ \cos\theta\sin\psi&\cos\phi\cos\psi+\sin\phi\sin\theta\sin\psi&-\sin\phi\cos% \psi+\cos\phi\sin\theta\sin\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\\ \end{bmatrix}$

NTCIR12-MathWiki-14rate: 2

$\cos c=\cos a\,\cos b+\sin a\,\sin b\,\cos\gamma.$

NTCIR12-MathWiki-14rate: 0

$\mathbf{\begin{bmatrix}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{bmatrix}=\begin{bmatrix}\frac{1}{a}&-\frac{\cos(\gamma)}{a\sin(\gamma)}&% \frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)}\\ 0&\frac{1}{b\sin(\gamma)}&\frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(% \gamma)}\\ 0&0&\frac{\sin(\gamma)}{cv}\\ \end{bmatrix}}\begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}$

NTCIR12-MathWiki-14rate: 0

$\sigma_{2}=\cos\psi\,d\theta+\sin\psi\sin\theta\,d\phi$

NTCIR12-MathWiki-14rate: 1

$\cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(% \varphi-\varphi^{\prime}).$

NTCIR12-MathWiki-14rate: 2

$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,$

NTCIR12-MathWiki-14rate: 1

$\sin\left(a+b\right)=\sin a\cos b+\cos a\sin b$

NTCIR12-MathWiki-14rate: 4

$\cos\alpha=-\cos\beta\cos\gamma+\sin\beta\sin\gamma\cosh\frac{a}{k},\,$

NTCIR12-MathWiki-14rate: 3

$\cos C=-\cos A\cos B+\sin A\sin B\cosh c,$

NTCIR12-MathWiki-14rate: 0

$\gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0$

NTCIR12-MathWiki-14rate: 3

$\cos A=-\cos B\cos C+\sin B\sin C\cosh a.\,$

NTCIR12-MathWiki-14rate: 0

$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$

NTCIR12-MathWiki-14rate: 0

$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

NTCIR12-MathWiki-14rate: 0

$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$

NTCIR12-MathWiki-14rate: 0

$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

NTCIR12-MathWiki-14rate: 0

$\begin{bmatrix}\cos(\alpha)\cos(\gamma)-\sin(\alpha)\sin(\beta)\sin(\gamma)&-% \sin(\alpha)\cos(\beta)&-\cos(\alpha)\sin(\gamma)-\sin(\alpha)\sin(\beta)\cos(% \gamma)\\ \cos(\alpha)\sin(\beta)\sin(\gamma)+\sin(\alpha)\cos(\gamma)&\cos(\alpha)\cos(% \beta)&\cos(\alpha)\sin(\beta)\cos(\gamma)-\sin(\alpha)\sin(\gamma)\\ \cos(\beta)\sin(\gamma)&-\sin(\beta)&\cos(\beta)\cos(\gamma)\end{bmatrix}$

NTCIR12-MathWiki-14rate: 0

$\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}=0$

NTCIR12-MathWiki-14rate: 0

$\begin{aligned}\displaystyle p_{i}&\displaystyle=\cos\Omega\cos\omega-\sin% \Omega\cos i\sin\omega\\ \displaystyle p_{j}&\displaystyle=\sin\Omega\cos\omega+\cos\Omega\cos i\sin% \omega\\ \displaystyle p_{k}&\displaystyle=\sin i\sin\omega\\ \displaystyle q_{i}&\displaystyle=-\cos\Omega\sin\omega-\sin\Omega\cos i\cos% \omega\\ \displaystyle q_{j}&\displaystyle=-\sin\Omega\sin\omega+\cos\Omega\cos i\cos% \omega\\ \displaystyle q_{k}&\displaystyle=\sin i\cos\omega\\ \displaystyle w_{i}&\displaystyle=\sin i\sin\Omega\\ \displaystyle w_{j}&\displaystyle=-\sin i\cos\Omega\\ \displaystyle w_{k}&\displaystyle=\cos i\end{aligned}$

NTCIR12-MathWiki-14rate: 0

$\cos(A+B)=\cos(A)\cos(B)+\sin(A)\sin(B)\,$

NTCIR12-MathWiki-14rate: 0

$\frac{\partial u}{\partial\varphi}=-\frac{\partial u}{\partial x}r\sin\varphi+% \frac{\partial u}{\partial y}r\cos\varphi=-y\frac{\partial u}{\partial x}+x% \frac{\partial u}{\partial y}.$

NTCIR12-MathWiki-14rate: 0

$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$

NTCIR12-MathWiki-14rate: 0

$\cos(\alpha+\beta)+\cos(\alpha-\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin% \beta+\cos\alpha\cos\beta+\sin\alpha\sin\beta=2\cos\alpha\cos\beta$

NTCIR12-MathWiki-14rate: 0

$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$

NTCIR12-MathWiki-14rate: 0

$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\,$

NTCIR12-MathWiki-14rate: 2

$\cos a=\cos b\cos c+\sin b\sin c\cos\alpha$

NTCIR12-MathWiki-14rate: 1

$\sin(\alpha+\beta)\sin(\beta+\gamma)=\sin\alpha\sin\gamma+\sin\beta\sin(\alpha% +\beta+\gamma)$

NTCIR12-MathWiki-14rate: 0

$\cos(x+y)=\cos{x}\cos{y}-\sin{x}\sin{y}$

NTCIR12-MathWiki-14rate: 1

$\sin\alpha\sin\beta\cos\beta\cos\gamma+\sin\alpha\cos^{2}\beta\sin\gamma+\cos% \alpha\sin^{2}\beta\cos\gamma+\cos\alpha\sin\beta\cos\beta\sin\gamma$

NTCIR12-MathWiki-14rate: 0

$\sin(x+y)=\sin x\,\cos y\ +\ \sin y\,\cos x$

NTCIR12-MathWiki-14rate: 0

$\cos(\theta-\alpha)=\cos(\theta)\cos(\alpha)+\sin(\theta)\sin(\alpha)$

NTCIR12-MathWiki-14rate: 0

$\mathbf{R}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\\ \end{pmatrix}\begin{pmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{pmatrix}$

NTCIR12-MathWiki-14rate: 0

$\sin\alpha_{\mathrm{s}}=\cos h\cos\delta\cos\varphi+\sin\delta\sin\varphi$

NTCIR12-MathWiki-14rate: 1

$a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}% \right),$

NTCIR12-MathWiki-14rate: 1

$c=\arccos\left(\cos a\cos b+\sin a\sin b\cos\gamma\right)$

NTCIR12-MathWiki-14rate: 1

$a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)$

NTCIR12-MathWiki-14rate: 0

$\cos(\theta^{\prime}-\theta)=\cos\theta^{\prime}\cos\theta+\sin\theta\sin% \theta^{\prime}$

NTCIR12-MathWiki-14rate: 4

$\cos(A)=-\cos(B)\cos(C)+\sin(B)\sin(C)\cos(a)\,$

NTCIR12-MathWiki-14rate: 3

$\cos A=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,$

NTCIR12-MathWiki-14rate: 3

$\cos B=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,$

NTCIR12-MathWiki-14rate: 0

$C\cos{\mu}+L\sin{\mu}+T(\sin{\alpha}\sin{\mu}+\cos{\alpha}\cos{\mu}\sin{\beta}% )=\frac{W}{g}\frac{(V\cos{\gamma})^{2}}{R}\quad(x_{E}\,\text{-}y_{E}\,\text{ % plane radial direction})$

NTCIR12-MathWiki-14rate: 1

$\sin G=\cos L\cos D\cos R-\sin L\sin R=-\cos L\cos D\sin I+\sin L\cos I$

NTCIR12-MathWiki-14rate: 0

$\cos(x-y)=\cos x\cos y+\sin x\sin y\,$

NTCIR12-MathWiki-14rate: 0

$\cos\left(x-y\right)=\cos x\cos y+\sin x\sin y.\,$

NTCIR12-MathWiki-14rate: 0

$\cos(A\pm B)=\cos A\ \cos B\mp\sin A\ \sin B$

NTCIR12-MathWiki-14rate: 1

$\mathbf{\hat{r}}=\sin\theta\cos\varphi\mathbf{\hat{x}}+\sin\theta\sin\varphi% \mathbf{\hat{y}}+\cos\theta\mathbf{\hat{z}}$

NTCIR12-MathWiki-14rate: 0

$\hat{\mathbf{x}}_{2}=[-\cos\beta\sin\gamma_{2}\,,\,\cos\gamma_{2}\,,\,\sin% \beta\sin\gamma_{2}]$

NTCIR12-MathWiki-14rate: 0

$\alpha_{2}=\arctan\left(\frac{\sin\alpha}{-\sin U_{1}\sin\sigma+\cos U_{1}\cos% \sigma\cos\alpha_{1}}\right)\,$

NTCIR12-MathWiki-14rate: 0

$\gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0$

NTCIR12-MathWiki-14rate: 0

$\begin{array}[]{lcl}\hat{\mathcal{P}}_{1}&=&\,i\left({\cos\gamma\over\sin\beta% }{\partial\over\partial\alpha}-\sin\gamma{\partial\over\partial\beta}-\cot% \beta\cos\gamma{\partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{2}&=&\,i\left(-{\sin\gamma\over\sin\beta}{\partial\over% \partial\alpha}-\cos\gamma{\partial\over\partial\beta}+\cot\beta\sin\gamma{% \partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{3}&=&-i{\partial\over\partial\gamma},\\ \end{array}$

NTCIR12-MathWiki-15rate: 0

$x,y\in A^{\ast}$

NTCIR12-MathWiki-15rate: 2

$x\in V\land\exists y(y\in x)\rightarrow\exists y(y\in x\land\lnot\exists z(z% \in y\land z\in x)).$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in V,$

NTCIR12-MathWiki-15rate: 0

$A=\{x\in X|(\exists y\in Y)\langle x,y\rangle\in B\}.$

NTCIR12-MathWiki-15rate: 1

$\mu(x+y)+\mu(x)+\mu(y)\equiv\lambda(x,y)\;\;(\mathop{{\rm mod}}2)\;\forall\,x,% y\in H_{k}(M;\mathbb{Z}_{2})$

NTCIR12-MathWiki-15rate: 1

$\forall x\,\forall y\,\exists w\,\forall z\,[z\in w\leftrightarrow(z\in xz=y)].$

NTCIR12-MathWiki-15rate: 1

$\forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge\phi(z))$

NTCIR12-MathWiki-15rate: 1

$\forall w_{1},\ldots,w_{n}\,[(\forall x\,\exists\,y\phi(x,y,w_{1},\ldots,w_{n}% ))\Rightarrow\forall A\,\exists B\,\forall x\in A\,\exists y\in B\,\phi(x,y,w_% {1},\ldots,w_{n})]$

NTCIR12-MathWiki-15rate: 0

$\forall w_{1},\ldots,w_{n}\,\forall A\,\exists B\,\forall x\in A\,[\exists y% \phi(x,y,w_{1},\ldots,w_{n})\Rightarrow\exists y\in B\,\phi(x,y,w_{1},\ldots,w% _{n})]$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in A:\|x\,y\|\ \leq\|x\|\,\|y\|$

NTCIR12-MathWiki-15rate: 1

$\forall x,y\in A,f(x)=f(y)\Rightarrow x=y.$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in A,x\neq y\Rightarrow f(x)\neq f(y).$

NTCIR12-MathWiki-15rate: 0

$\forall xy\,[x\neq y\rightarrow\exists z\neq y\,C(x;yz)].$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in G$

NTCIR12-MathWiki-15rate: 1

$\forall A\;([\forall x\in A\;\exists y\;\phi(x,y)]\to\exists B\;\forall x\in A% \;\exists y\in B\;\phi(x,y))$

NTCIR12-MathWiki-15rate: 0

$\forall a((\forall x\in a\;\exists y\;\phi(x,y))\to\exists b\;(\forall x\in a% \;\exists y\in b\;\phi(x,y))\wedge(\forall y\in b\;\exists x\in a\;\phi(x,y)))$

NTCIR12-MathWiki-15rate: 0

$L(x)=\sum_{i}l_{i}x^{p^{i}},l_{i}\in\mathrm{GF}(p^{m}).$

NTCIR12-MathWiki-15rate: 0

$x,y\in A$

NTCIR12-MathWiki-15rate: 0

$\forall x\Big(\forall y(y\in x\rightarrow P[y])\rightarrow P[x]\Big)% \rightarrow\forall x\,P[x]$

NTCIR12-MathWiki-15rate: 1

$\forall a,b,c\in X\,(b\,R\,a\land c\,R\,a\to b\,R\,c).$

NTCIR12-MathWiki-15rate: 0

$\exists_{x}\forall_{y}(x\neq y\Rightarrow G(x,y)).$

NTCIR12-MathWiki-15rate: 1

$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow\forall z% (x\in z\Leftrightarrow y\in z)]$

NTCIR12-MathWiki-15rate: 0

$\forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow(C\in A\rightarrow D% \in B).$

NTCIR12-MathWiki-15rate: 1

$\Box A=\{x\in F;\,\forall y\in F\,(x\,R\,y\to y\in A)\}$

NTCIR12-MathWiki-15rate: 1

$\forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\rightarrow x=y].$

NTCIR12-MathWiki-15rate: 2

$\forall z\exists y\forall x[x\in y\leftrightarrow(x\in z\land\phi(x))].$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in A$

NTCIR12-MathWiki-15rate: 1

$\forall x,y,z\,\left(xFy\wedge xFz\to y=z\right)$

NTCIR12-MathWiki-15rate: 1

$\forall x,y,z\,(xRy\wedge yRz\rightarrow xRz)$

NTCIR12-MathWiki-15rate: 2

$\forall n\in\mathbb{Z}(\exists m\in\mathbb{Z}[m>n\wedge P(m)])$

NTCIR12-MathWiki-15rate: 1

$\forall^{\mathrm{st}}x\,\exists^{\mathrm{st}}y\,\forall^{\mathrm{st}}t\,(t\in y% \leftrightarrow(t\in x\land\phi(t,u_{1},\dots,u_{n})))$

NTCIR12-MathWiki-15rate: 0

$(g\cdot f)(x):=f(g^{-1}x)\qquad\forall x\in V,g\in G,f\in k[V].$

NTCIR12-MathWiki-15rate: 0

$z\in L\implies\Pr\nolimits_{x}[\exists y.\phi(x,y,z)]\geq\tfrac{2}{3}$

NTCIR12-MathWiki-15rate: 0

$x\neq z\to M[x:=y][z]=M[z]$

NTCIR12-MathWiki-15rate: 0

$(\forall z:z\not\in FV(y)z\not\in BV(b))\to\operatorname{beta-redex}[\lambda x% .b\ y]=b[x:=y]$

NTCIR12-MathWiki-15rate: 0

$\iff x=y\and(L\ z)[x:=y]$

NTCIR12-MathWiki-15rate: 0

$l_{x}(y)=xy,x\in\mathfrak{g},y\in U(\mathfrak{g})$

NTCIR12-MathWiki-15rate: 1

$\forall\epsilon>0\ \exists\delta>0\ \forall y\in X\ \left[d(x,y)<\delta% \Rightarrow\forall n\in\mathbf{N}\ d\left(f^{n}(x),f^{n}(y)\right)<\epsilon% \right].$

NTCIR12-MathWiki-15rate: 1

$\forall X\subseteq U_{p}[CUM_{p}(X)\Leftrightarrow\exists x,y[X(x)\,\wedge\,X(% y)\,\wedge\,\neg(x=y)]\;\wedge\;\forall x,y[X(x)\,\wedge\,X(y)\Rightarrow X(x% \,\oplus\,y)]]$

NTCIR12-MathWiki-15rate: 1

$Pxy\leftrightarrow\forall z[Ozx\rightarrow Ozy].$

NTCIR12-MathWiki-15rate: 1

$\forall x\exist y[Pyx\and(Czy\rightarrow Ozx)\and\lnot(Pxy\and(Czx\rightarrow Ozy% ))].$

NTCIR12-MathWiki-15rate: 1

$\forall x,y.P(x,y)\vee Q(f(x))$

NTCIR12-MathWiki-15rate: 0

$C:=\{b\in B:\exists\,p(x)\in A[x]\,,\hbox{ which is monic and such that }p(b)=% 0\}\,.$

NTCIR12-MathWiki-15rate: 1

$\forall a\,\forall s\,[(Ma\and\forall x\,[x\in s\leftrightarrow\exists y\,(x% \in y\and y\in a)])\rightarrow Ms].$

NTCIR12-MathWiki-15rate: 1

$\exists y[My\and\varnothing\in y\and\forall z(z\in y\rightarrow\exists x[x\in y% \and\forall w(w\in x\leftrightarrow[w=zw\in z])])].$

NTCIR12-MathWiki-15rate: 0

$x,y\in A$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in A\qquad\|xy\|\leq\|x\|\|y\|$

NTCIR12-MathWiki-15rate: 1

$(\forall x)Eq(x,x)\wedge(\forall x,y,z)[Eq(x,y)\rightarrow(Eq(x,z)\rightarrow Eq% (y,z))]$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in N$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in N$

NTCIR12-MathWiki-15rate: 1

$\forall x,y\in Nx<y\Rightarrow\exists z\in Nx+z=y$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in N$

NTCIR12-MathWiki-15rate: 0

$\forall x,y:\neg\;(x<y\;\wedge\;y<x)$

NTCIR12-MathWiki-15rate: 0

$x,y\in A$

NTCIR12-MathWiki-15rate: 0

$g^{M}_{p}(u,v)=g^{N}_{f(p)}(df(u),df(v))\qquad\forall p\in M,\forall u,v\in T_% {p}M.$

NTCIR12-MathWiki-15rate: 0

$\exists y\forall x(x\in y\iff P(x))$

NTCIR12-MathWiki-15rate: 1

$\forall x[\forall y(Rxy\rightarrow\exists z[Ryz])\rightarrow\exists y(Rxy% \wedge\forall z[Ryz\rightarrow z=y])]$

NTCIR12-MathWiki-15rate: 0

$x,y\in A$

NTCIR12-MathWiki-15rate: 4

$\forall x,y\in A\;[x\neq y\rightarrow\neg\exists z\in X\;[z\leq x\land z\leq y% ]].$

NTCIR12-MathWiki-15rate: 1

$\forall x,y[x\neq y\rightarrow[xRy\vee yRx]]$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in A,f(x+y)\leq f(x)+f(y).$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in G$

NTCIR12-MathWiki-15rate: 0

$u\equiv_{S}v\Leftrightarrow\forall x,y\in M(xuy\in S\Leftrightarrow xvy\in S).$

NTCIR12-MathWiki-15rate: 1

$\forall r\forall x[Fxr\leftrightarrow[\forall y(y\in x\rightarrow Byr)\and% \lnot Bxr]]\,.$

NTCIR12-MathWiki-15rate: 0

$\exist r\forall y[A(y)\rightarrow Byr]\rightarrow\exist x\forall y[y\in x% \leftrightarrow A(y)]\,.$

NTCIR12-MathWiki-15rate: 0

$\forall x\exists y[x\in y\wedge\forall z\in y(\mathcal{P}(z)\subseteq y\wedge% \mathcal{P}(z)\in y)\wedge\forall z\in\mathcal{P}(y)(\neg z\approx y\to z\in y)]$

NTCIR12-MathWiki-15rate: 0

$x,y\in A_{1}$

NTCIR12-MathWiki-15rate: 1

$\forall n\in\mathbb{N}\ \exists A\subseteq\mathbb{N}\ \forall x\in\mathbb{N}\ % [x\in A\,\text{ iff }x\leq n].$

NTCIR12-MathWiki-15rate: 0

$\langle Tx\mid y\rangle_{2}=\left\langle x\mid T^{*}y\right\rangle_{1},\qquad x% \in D(T).$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in U,~{}x\neq y$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in U,~{}x\neq y$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in U,~{}x\neq y$

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$\forall x\forall y\exists z\forall w[w\in z\leftrightarrow(w=xw=y)].$

NTCIR12-MathWiki-15rate: 0

$\forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S$

NTCIR12-MathWiki-15rate: 0

$x<y\rightarrow\exists z[x<z<y].$

NTCIR12-MathWiki-15rate: 0

$\forall z[z<x\rightarrow z<y]\rightarrow x\leq y.$

NTCIR12-MathWiki-15rate: 0

$\forall x\forall y\exists z(x\in z\land y\in z).$

NTCIR12-MathWiki-15rate: 1

$\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].$

NTCIR12-MathWiki-16rate: 3

$\lambda\text{piv}=\sqrt{\frac{\int e(\lambda)\lambda d\lambda}{\int e(\lambda)% \lambda^{-1}d\lambda}}$

NTCIR12-MathWiki-16rate: 0

$a(t)=e^{\int_{0}^{t}\delta_{u}\,du}$

NTCIR12-MathWiki-16rate: 1

$\sigma_{y}^{2}(\tau)=\int_{0}^{\infty}S_{y}(f)\frac{2\sin^{4}\pi\tau f}{(\pi% \tau f)^{2}}\,df$

NTCIR12-MathWiki-16rate: 0

$x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[a_{n}(t)\cos\left(2\pi n% \int_{0}^{t}f_{0}(\tau)\,d\tau\right)-b_{n}(t)\sin\left(2\pi n\int_{0}^{t}f_{0% }(\tau)\,d\tau\right)\right]$

NTCIR12-MathWiki-16rate: 0

$T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},$

NTCIR12-MathWiki-16rate: 0

$M=\frac{1}{1-\int_{0}^{L}\alpha(x)\,dx}$

NTCIR12-MathWiki-16rate: 4

$D4\sigma=4\sigma=4\sqrt{\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(% x,y)(x-\bar{x})^{2}\,dx\,dy}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x% ,y)\,dx\,dy}}$

NTCIR12-MathWiki-16rate: 0

$T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},$

NTCIR12-MathWiki-16rate: 0

$J_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(\alpha\tau-x\sin\tau)\,d\tau-% \frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}e^{-x\sinh(t)-\alpha t}\,dt.$

NTCIR12-MathWiki-16rate: 1

$p(t)=\int_{0}^{\infty}h(\tau)q(t-\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 0

$\overline{\left[\left(\tau_{s}-\bar{\tau}_{s}\right)^{2}\right]}^{\frac{1}{2}}% =1.4\sqrt{s},$

NTCIR12-MathWiki-16rate: 1

$=\sqrt{\frac{\sum{(y_{i}-mx_{i}-b)}^{2}}{n-2}}$

NTCIR12-MathWiki-16rate: 1

$f(t)=\int_{0}^{\infty}h(\tau)s(t-\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 2

$C=\frac{\int xg(x)\;dx}{\int g(x)\;dx}$

NTCIR12-MathWiki-16rate: 2

$C_{k}=\frac{\int zS_{k}(z)\;dz}{\int S_{k}(z)\;dz}$

NTCIR12-MathWiki-16rate: 0

$\lim_{\lambda\to\infty}\int_{0}^{\lambda}\left(1-\frac{x}{\lambda}\right)^{% \alpha}f(x)\,dx$

NTCIR12-MathWiki-16rate: 2

$s_{p}=\sqrt{\frac{p\,(1-p)}{n}}$

NTCIR12-MathWiki-16rate: 0

$a(n)=e^{\int_{0}^{n}\delta_{t}\,dt}\ ,$

NTCIR12-MathWiki-16rate: 2

$\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}f(\tau)\,g(t-\tau)\,d\tau$

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$=\int_{-\infty}^{\infty}f(t-\tau)\,g(\tau)\,d\tau.$

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$\tau_{\,\text{rms}}=\sqrt{\frac{\int_{0}^{\infty}(\tau-\overline{\tau})^{2}A_{% c}(\tau)d\tau}{\int_{0}^{\infty}A_{c}(\tau)d\tau}}$

NTCIR12-MathWiki-16rate: 0

$=\int\limits_{-\infty}^{\infty}f(\tau)\delta(\tau-(t-T))\,d\tau$

NTCIR12-MathWiki-16rate: 1

$v_{C}(t)=V_{0}+{1\over C}\int_{0}^{t}i_{C}(\tau)\,d\tau\iff i_{L}(t)=I_{0}+{1% \over L}\int_{0}^{t}v_{L}(\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 2

$=\int_{0}^{\infty}G(\tau)F(t-\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 0

$=\int_{-\infty}^{t}G(t-\tau)F(\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 1

$\overline{\left(A_{t}-\overline{A}\right)^{2}}\equiv\lim_{\tau\to\infty}\frac{% 1}{\tau}\int_{0}^{\tau}\left(A_{t}-\overline{A}\right)^{2}dt,$

NTCIR12-MathWiki-16rate: 0

$\begin{aligned}\displaystyle x&\displaystyle=\int_{0}^{L}\cos\theta\,ds\\ &\displaystyle=\int_{0}^{L}\cos\left[(as)^{2}\right]ds\end{aligned}$

NTCIR12-MathWiki-16rate: 0

$Q=\int_{0}^{t}I(\tau)\ d\tau$

NTCIR12-MathWiki-16rate: 1

$\Delta(x)=\int_{0}^{\infty}d\tau\int DXe^{-\int_{0}^{\tau}(\dot{x}^{2}/2+m^{2}% )d\tau^{\prime}}$

NTCIR12-MathWiki-16rate: 0

$e^{-\int_{0}^{t}V(x(\tau))\,d\tau}$

NTCIR12-MathWiki-16rate: 1

$f(t)=2\int_{0}^{\infty}\int_{-\infty}^{\infty}f(\tau)\cos 2\pi\lambda(\tau-t)d% \tau d\lambda.$

NTCIR12-MathWiki-16rate: 1

$\begin{aligned}\displaystyle(J^{\alpha})(J^{\beta}f)(x)&\displaystyle=\frac{1}% {\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}(J^{\beta}f)(t)\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}\int_{0}^{t}(x% -t)^{\alpha-1}(t-s)^{\beta-1}f(s)\;ds\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}f(s)\left(\int% _{s}^{x}(x-t)^{\alpha-1}(t-s)^{\beta-1}\;dt\right)ds\end{aligned}$

NTCIR12-MathWiki-16rate: 0

$=A_{c}\cos\left(2\pi f_{c}t+2\pi f_{\Delta}\int_{0}^{t}x_{m}(\tau)d\tau\right)$

NTCIR12-MathWiki-16rate: 1

$\int_{0}^{t}x_{m}(\tau)d\tau=\frac{A_{m}\cos(2\pi f_{m}t)}{2\pi f_{m}}\,$

NTCIR12-MathWiki-16rate: 4

$(\Delta x)^{2}=\frac{\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)f^{*}(x)\,dx}{\int_% {-\infty}^{\infty}f(x)f^{*}(x)\,dx}$

NTCIR12-MathWiki-16rate: 4

$(\Delta k)^{2}=\frac{\int_{-\infty}^{\infty}(k-k_{0})^{2}F(k)F^{*}(k)\,dk}{% \int_{-\infty}^{\infty}F(k)F^{*}(k)\,dk}$

NTCIR12-MathWiki-16rate: 1

$GEH=\sqrt{\frac{2(M-C)^{2}}{M+C}}$

NTCIR12-MathWiki-16rate: 4

$T=\frac{\int_{x=0}^{\infty}x\ell(x)m(x)\,\mathrm{d}x}{\int_{x=0}^{\infty}\ell(% x)m(x)\,\mathrm{d}x}$

NTCIR12-MathWiki-16rate: 2

$GWP\left(x\right)=\frac{\int_{0}^{TH}a_{x}\cdot\left[x(t)\right]dt}{\int_{0}^{% TH}a_{r}\cdot\left[r(t)\right]dt}$

NTCIR12-MathWiki-16rate: 1

$q_{xx}=\frac{\sum(x-\bar{x})^{2}I(x,y)}{\sum I(x,y)}$

NTCIR12-MathWiki-16rate: 1

$Y(t)=\chi\text{i}X(t)+\int_{0}^{\infty}\Phi\text{d}(\tau)X(t-\tau)\,\mathrm{d}\tau,$

NTCIR12-MathWiki-16rate: 0

$D_{\gamma}(\gamma(b)||\gamma(a))=\int_{a}^{b}(b-s)g_{\gamma}(s)ds$

NTCIR12-MathWiki-16rate: 3

$\delta(x)=\frac{\int_{-\infty}^{\infty}{\theta f(x_{1}-\theta,\dots,x_{n}-% \theta)d\theta}}{\int_{-\infty}^{\infty}{f(x_{1}-\theta,\dots,x_{n}-\theta)d% \theta}}.$

NTCIR12-MathWiki-16rate: 3

$M\text{an}\text{aniso}=M\text{s}\frac{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(% \theta)\cos(\theta)\,d\theta}{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(\theta)\,% d\theta}$

NTCIR12-MathWiki-16rate: 1

$\begin{aligned}\displaystyle\mathbf{E}[G|H]&\displaystyle=\int^{T}_{0}k(t)% \mathbf{E}[x(t)|H]dt=0\\ \displaystyle\mathbf{E}[G|K]&\displaystyle=\int^{T}_{0}k(t)\mathbf{E}[x(t)|K]% dt=\int^{T}_{0}k(t)S(t)dt\equiv\rho\\ \displaystyle\mathbf{E}[G^{2}|H]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )R_{N}(t,s)dtds=\int^{T}_{0}k(t)\left(\int^{T}_{0}k(s)R_{N}(t,s)ds\right)=\int% ^{T}_{0}k(t)S(t)dt=\rho\\ \displaystyle\,\text{Var}[G|H]&\displaystyle=\mathbf{E}[G^{2}|H]-(\mathbf{E}[G% |H])^{2}=\rho\\ \displaystyle\mathbf{E}[G^{2}|K]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )\mathbf{E}[x(t)x(s)]dtds=\int^{T}_{0}\int^{T}_{0}k(t)k(s)(R_{N}(t,s)+S(t)S(s)% )dtds=\rho+\rho^{2}\\ \displaystyle\,\text{Var}[G|K]&\displaystyle=\mathbf{E}[G^{2}|K]-(\mathbf{E}[G% |K])^{2}=\rho+\rho^{2}-\rho^{2}=\rho\end{aligned}$

NTCIR12-MathWiki-16rate: 1

$\frac{d\vec{P}}{dt}=\int^{t}_{0}\mathbf{A}(t-\tau)\vec{P}(\tau)d\tau$

NTCIR12-MathWiki-16rate: 3

$\alpha_{\mathrm{sun}}=\displaystyle\frac{\int_{0}^{\infty}\alpha_{\lambda}I_{% \lambda\mathrm{sun}}(\lambda)\,d\lambda}{\int_{0}^{\infty}I_{\lambda\mathrm{% sun}}(\lambda)\,d\lambda}$

NTCIR12-MathWiki-16rate: 2

$\langle v_{e}\rangle=\frac{\int_{v_{e0}}^{\infty}f(v_{x})\,v_{x}\,dv_{x}}{\int% _{-\infty}^{\infty}f(v_{x})\,dv_{x}}$

NTCIR12-MathWiki-16rate: 2

$y(t)=\int_{0}^{T}x(t-\tau)\,h(\tau)\,d\tau$

NTCIR12-MathWiki-16rate: 1

$\int_{0}^{t}\mathrm{ROI}(\tau)\,d\tau=(-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}% \mathbf{Q}+V_{p})\int_{0}^{t}C_{p}(\tau)\,d\tau+\mathbf{U}_{n}^{T}\mathbf{K}^{% -1}\mathbf{A}$

NTCIR12-MathWiki-16rate: 0

${{\int_{0}^{t}\mathrm{ROI}(\tau)d\tau}\over\mathrm{ROI}(t)}$

NTCIR12-MathWiki-16rate: 1

$\frac{\int^{\infty}_{0}y_{\lambda}J_{\lambda}d\lambda}{\int^{\infty}_{0}J_{% \lambda}d\lambda},$

NTCIR12-MathWiki-16rate: 0

$\Omega_{1}=\int_{0}^{t}A(\tau)d\tau$

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$g(s)=\sqrt{2/\pi}\int_{0}^{\infty}(st)^{1/2}\,K_{\nu}(st)\,f(t)\;dt,$

NTCIR12-MathWiki-16rate: 0

$\begin{aligned}\displaystyle m(\varphi)&\displaystyle=\int_{0}^{\varphi}M(% \varphi)\,d\varphi=a(1-e^{2})\int_{0}^{\varphi}\bigl(1-e^{2}\sin^{2}\varphi% \bigr)^{-3/2}\,d\varphi.\end{aligned}$

NTCIR12-MathWiki-16rate: 1

$\mu_{n}=\int_{-\infty}^{\infty}(x-c)^{n}\,f(x)\,dx.$

NTCIR12-MathWiki-16rate: 1

$\nu_{r}=\int(x-\mu)^{r}f(x)\,dx$

NTCIR12-MathWiki-16rate: 1

$S(\tau_{B})-S(\tau_{A})=\int_{A}^{B}dS=\int_{\tau_{A}}^{\tau_{B}}\frac{dS}{d% \tau}d\tau=\int_{\tau_{A}}^{\tau_{B}}\frac{S(\tau+d\tau)-S(\tau)}{(\tau+d\tau)% -\tau}d\tau$

NTCIR12-MathWiki-16rate: 1

$\sqrt{\frac{\Sigma(x-\bar{x})^{2}}{n}}=\sqrt{\frac{\Sigma\left[x-\left(\Sigma x% \right)/n\right]^{2}}{n}}\ ,$

NTCIR12-MathWiki-16rate: 3

$\Omega(r)=\frac{\int_{r}^{\infty}(1-F(x))\,dx}{\int_{-\infty}^{r}F(x)dx}$

NTCIR12-MathWiki-16rate: 2

$\frac{1}{\kappa}=\frac{\int_{0}^{\infty}\kappa_{\nu}^{-1}u(\nu,T)d\nu}{\int_{0% }^{\infty}u(\nu,T)d\nu}$

NTCIR12-MathWiki-16rate: 4

$\frac{1}{\kappa}=\frac{\int_{0}^{\infty}(\kappa_{\nu,{\rm es}}+\kappa_{\nu,{% \rm ff}})^{-1}u(\nu,T)d\nu}{\int_{0}^{\infty}u(\nu,T)d\nu}$

NTCIR12-MathWiki-16rate: 1

$\mathrm{Var}(x)=\int_{-\infty}^{\infty}(x-\langle x\rangle)^{2}P_{n}(x)\,dx=% \frac{L^{2}}{12}\left(1-\frac{6}{n^{2}\pi^{2}}\right)$

NTCIR12-MathWiki-16rate: 1

$G(\vec{R},\vec{R}^{\prime};N)\equiv\frac{\displaystyle\int_{\vec{R}_{0}=\vec{R% }^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}(n)\exp\left[-\frac{3}{2l^{% 2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec{R}_{n}}{\partial n}% \right)^{2}-\beta\displaystyle\int_{0}^{N}duU_{e}[\vec{R}(n)]\right]}{% \displaystyle\int d\vec{R}^{\prime}\displaystyle\int d\vec{R}\displaystyle\int% _{\vec{R}_{0}=\vec{R}^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}_{n}% \exp\left[-\frac{3}{2l^{2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec% {R}_{n}}{\partial n}\right)^{2}\right]}$

NTCIR12-MathWiki-16rate: 1

$R(t)=K\int_{0}^{t}C_{p}(\tau)\,d\tau+V_{0}C_{p}(t)$

NTCIR12-MathWiki-16rate: 1

$u=\sqrt{\frac{2(p_{t}-p_{s})}{\rho}}$

NTCIR12-MathWiki-16rate: 2

$d=\sqrt{\int_{-\infty}^{t}(t-r)^{2}f(r)\,dr}$

NTCIR12-MathWiki-16rate: 3

$f_{X\mid Y=y}(x)={f_{X}(x)L_{X\mid Y=y}(x)\over{\int_{-\infty}^{\infty}f_{X}(x% )L_{X\mid Y=y}(x)\,dx}}$

NTCIR12-MathWiki-16rate: 0

$I_{\mbox{rms}~{}}$

NTCIR12-MathWiki-16rate: 1

$\int_{-x}^{x}(x^{2}-y^{2})^{n}\cos(y)\,dy.$

NTCIR12-MathWiki-16rate: 1

$\begin{aligned}\displaystyle J_{n}\left(\frac{\pi}{2}\right)&\displaystyle=% \int_{-\pi/2}^{\pi/2}\left(\frac{\pi^{2}}{4}-y^{2}\right)^{n}\cos(y)\,dy\\ &\displaystyle=\int_{0}^{\pi}\left(\frac{\pi^{2}}{4}-\left(y-\frac{\pi}{2}% \right)^{2}\right)^{n}\cos\left(y-\frac{\pi}{2}\right)\,dy\\ &\displaystyle=\int_{0}^{\pi}y^{n}(\pi-y)^{n}\sin(y)\,dy\\ &\displaystyle=\frac{n!}{b^{n}}\int_{0}^{\pi}f(x)\sin(x)\,dx.\end{aligned}$

NTCIR12-MathWiki-16rate: 1

$d_{k}=\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx$

NTCIR12-MathWiki-16rate: 0

$\frac{\partial}{\partial t}\rho_{I}(t)=-\frac{1}{\hbar^{2}}\sum_{m,n}\int_{0}^% {\infty}d\tau\biggl(C_{mn}(\tau)\Bigl[S_{m,I}(t),S_{n,I}(t-\tau)\rho_{I}(t)% \Bigr]-C_{mn}^{\ast}(\tau)\Bigl[S_{m,I}(t),\rho_{I}(t)S_{n,I}(t-\tau)\Bigr]\biggr)$

NTCIR12-MathWiki-16rate: 1

$\sigma^{2}=\int_{0}^{\infty}(t-\bar{t})^{2}\cdot E(t)\,dt$

NTCIR12-MathWiki-16rate: 2

$E(t)=\frac{C(t)}{\int_{0}^{\infty}C(t)\ dt}$

NTCIR12-MathWiki-16rate: 1

$\operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}(\hat{y}_{t}-y)^{2}}{n}}.$

NTCIR12-MathWiki-16rate: 0

$\int_{-\infty}^{\infty}w(\tau)\,d\tau=1.$

NTCIR12-MathWiki-16rate: 1

$I_{1}=\tau\int_{0}^{\infty}H(\mu-\tau x)\,\mathrm{d}x-\tau\int_{0}^{\infty}% \frac{H(\mu-\tau x)}{e^{x}+1}\,\mathrm{d}x\,$

NTCIR12-MathWiki-16rate: 2

$DR=\sqrt{\int_{-\infty}^{T}(T-r)^{2}f(r)\,dr}$

NTCIR12-MathWiki-16rate: 0

$\langle\tau\rangle\equiv\int_{0}^{\infty}dt\,e^{-\left({t/\tau_{K}}\right)^{% \beta}}={\tau_{K}\over\beta}\Gamma({1\over\beta})$

NTCIR12-MathWiki-16rate: 1

$\mathbf{H}_{\alpha}(x)=\frac{2\left(\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}% \Gamma\left(\alpha+\frac{1}{2}\right)}\int_{0}^{\frac{\pi}{2}}\sin(x\cos\tau)% \sin^{2\alpha}(\tau)d\tau.$

NTCIR12-MathWiki-16rate: 1

$\begin{aligned}\displaystyle A&\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\sqrt{% \left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}\,dt\\ &\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\,dt\\ &\displaystyle{}=4\pi.\end{aligned}$

NTCIR12-MathWiki-16rate: 0

$T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}=10^{-\sum_{i% =1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z},$

NTCIR12-MathWiki-16rate: 2

$\Delta t^{2}=\left[\int^{\Delta\tau}_{0}e^{\int^{\bar{\tau}}_{0}a(\tau^{\prime% })d\tau^{\prime}}\,d\bar{\tau}\right]\,\left[\int^{\Delta\tau}_{0}e^{-\int^{% \bar{\tau}}_{0}a(\tau^{\prime})d\tau^{\prime}}\,d\bar{\tau}\right],$

NTCIR12-MathWiki-16rate: 1

$\begin{aligned}\displaystyle(3)\quad\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d% }t&\displaystyle=\int_{0}^{\delta}e^{-xt}t^{\lambda}g(t)\,\mathrm{d}t\\ &\displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)}{n!}\int_{0}^{\delta}t^{\lambda+% n}e^{-xt}\,\mathrm{d}t+\frac{1}{(N+1)!}\int_{0}^{\delta}g^{(N+1)}(t^{*})\,t^{% \lambda+N+1}e^{-xt}\,\mathrm{d}t.\end{aligned}$

NTCIR12-MathWiki-16rate: 2

$s=\sqrt{\frac{\sum_{i=1}^{n}{\left(x_{i}-\bar{x}\right)}^{2}}{n-1}}$

NTCIR12-MathWiki-16rate: 2

$v_{3}(t)=K_{1}\int_{t_{0}}^{t}i_{1}(\tau)d\tau+K_{2}\int_{t_{0}}^{t}i_{2}(\tau% )d\tau,$

NTCIR12-MathWiki-17rate: 0

$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1}{3}.$

NTCIR12-MathWiki-17rate: 0

$c=-\frac{1}{6}\cdot\frac{1}{2!}=-\frac{1}{12}.$

NTCIR12-MathWiki-17rate: 0

$\frac{1}{2}a_{0}-\frac{1}{4}\Delta a_{0}+\frac{1}{8}\Delta^{2}a_{0}-\cdots=% \frac{1}{2}-\frac{1}{4}.$

NTCIR12-MathWiki-17rate: 1

$\frac{a_{0}}{2}-\frac{\Delta a_{0}}{4}+\frac{\Delta^{2}a_{0}}{8}-\frac{\Delta^% {3}a_{0}}{16}+\cdots=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots.$

NTCIR12-MathWiki-17rate: 0

$\eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots$

NTCIR12-MathWiki-17rate: 0

$g=\left(1,0,1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},% \cdots\right)$

NTCIR12-MathWiki-17rate: 0

$\pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{10n}}\left(-\frac{2^{% 5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{% 10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!$

NTCIR12-MathWiki-17rate: 0

$\pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2% ^{5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}% }{10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!$

NTCIR12-MathWiki-17rate: 1

$\left(-\frac{3}{2}\right)+\left(-\frac{1}{2}\right)+\frac{1}{2}=\frac{3\left(-% \frac{3}{2}+\frac{1}{2}\right)}{2}=-\frac{3}{2}.$

NTCIR12-MathWiki-17rate: 1

$M_{O}=1-\tfrac{3}{5}=\tfrac{2}{5}$

NTCIR12-MathWiki-17rate: 0

$\scriptstyle B_{1}=-{1\over 2}$

NTCIR12-MathWiki-17rate: 0

$V=2(\frac{1}{r_{12}}-\frac{1}{r_{1a}}-\frac{1}{r_{1b}}-\frac{1}{r_{2a}}-\frac{% 1}{r_{2b}}+\frac{1}{r_{ab}})$

NTCIR12-MathWiki-17rate: 2

$(1-x^{2})^{1/3}=1-\frac{x^{2}}{3}-\frac{x^{4}}{9}-\frac{5x^{6}}{81}\cdots$

NTCIR12-MathWiki-17rate: 1

$C=\sum\frac{(-1)^{i}}{s_{i}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6}-\frac{1}{42% }+\frac{1}{1806}-\cdots\approx 0.64341054629.$

NTCIR12-MathWiki-17rate: 1

$1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots=\sum\limits_{n=1}^{\infty% }{(-1)^{n+1}\over n}$

NTCIR12-MathWiki-17rate: 1

${1\over 1}-{1\over 2}+{1\over 4}-{1\over 8}+{1\over 16}-{1\over 32}+\cdots={2% \over 3}.$

NTCIR12-MathWiki-17rate: 1

${1\over 1}+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+{1\over 6}+\cdots% \rightarrow\infty.$

NTCIR12-MathWiki-17rate: 1

$\psi(x)=\ln(x)-\frac{1}{2x}-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x% ^{6}}+\frac{1}{240x^{8}}-\frac{5}{660x^{10}}+\frac{691}{32760x^{12}}-\frac{1}{% 12x^{14}}+O\left(\frac{1}{x^{16}}\right)$

NTCIR12-MathWiki-17rate: 0

$\eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots$

NTCIR12-MathWiki-17rate: 1

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots=\sum_{n=1}^{\infty}% \frac{1}{n}.$

NTCIR12-MathWiki-17rate: 1

$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$

NTCIR12-MathWiki-17rate: 1

${\gamma=\sum_{k=2}^{\infty}(-1)^{k}\frac{\left\lfloor\log_{2}k\right\rfloor}{k% }=\frac{1}{2}-\frac{1}{3}+2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{% 7}\right)+3\left(\frac{1}{8}-\frac{1}{9}+\frac{1}{10}-\frac{1}{11}+\dots-\frac% {1}{15}\right)+\dots}$

NTCIR12-MathWiki-17rate: 1

$\tfrac{1}{2}(0-\tfrac{6}{4})=-\tfrac{3}{4}$

NTCIR12-MathWiki-17rate: 0

$B_{1}=-\frac{1}{2}$

NTCIR12-MathWiki-17rate: 1

$\frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots\;=\;1$

NTCIR12-MathWiki-17rate: 1

$\tfrac{2}{3}-\tfrac{1}{2}=\tfrac{4}{6}-\tfrac{3}{6}=\tfrac{1}{6}$

NTCIR12-MathWiki-17rate: 0

$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}=4-1% +\frac{1}{6}-\frac{1}{34}+\frac{16}{3145}-\frac{4}{4551}+\frac{1}{6601}-\frac{% 1}{38341}+-\cdots$

NTCIR12-MathWiki-17rate: 1

$\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)}=% \frac{1}{3}.$

NTCIR12-MathWiki-17rate: 4

$x-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}-\frac{1}{9}-\cdots=1$

NTCIR12-MathWiki-17rate: 0

$H_{\frac{1}{4}}=4-\tfrac{\pi}{2}-3\ln{2}$

NTCIR12-MathWiki-17rate: 1

$1\,-\,\frac{1}{2}\,+\,\frac{1}{3}\,-\,\frac{1}{4}\,+\,\frac{1}{5}\,-\,\cdots\;% =\;\ln 2.$

NTCIR12-MathWiki-17rate: 1

$\displaystyle 1\;\;+\;\;\frac{1}{2}\;\;+\;\;\frac{1}{3}\,+\,\frac{1}{4}\;\;+\;% \;\frac{1}{5}\,+\,\frac{1}{6}\,+\,\frac{1}{7}\,+\,\frac{1}{8}\;\;+\;\;\frac{1}% {9}\,+\,\cdots$

NTCIR12-MathWiki-17rate: 0

$H=\begin{bmatrix}1&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}\\ \frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}\\ \frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}\\ \frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}\\ \frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}&\frac{1}{9}\end{bmatrix}.$

NTCIR12-MathWiki-17rate: 1

$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$

NTCIR12-MathWiki-17rate: 0

$\frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots$

NTCIR12-MathWiki-17rate: 0

$\beta=-\frac{1}{2}$

NTCIR12-MathWiki-17rate: 0

$=\frac{\mu_{0}r^{2}N^{2}\pi}{l}\left(1-\frac{8w}{3\pi}+\frac{w^{2}}{2}-\frac{w% ^{4}}{4}+\frac{5w^{6}}{16}-\frac{35w^{8}}{64}+...\right)$

NTCIR12-MathWiki-17rate: 0

$\left([-1,1]+\frac{1}{2}\right)^{2}-\frac{1}{4}=\left[-\frac{1}{2},\frac{3}{2}% \right]^{2}-\frac{1}{4}=\left[0,\frac{9}{4}\right]-\frac{1}{4}=\left[-\frac{1}% {4},2\right]$

NTCIR12-MathWiki-17rate: 0

$\frac{1}{r_{\min}}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_{\max}}$

NTCIR12-MathWiki-17rate: 1

$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots$

NTCIR12-MathWiki-17rate: 0

$\frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots$

NTCIR12-MathWiki-17rate: 0

$W(e^{1+z})=1+\frac{z}{2}+\frac{z^{2}}{16}-\frac{z^{3}}{192}-\frac{z^{4}}{3072}% +\frac{13z^{5}}{61440}-\frac{47z^{6}}{1474560}-\frac{73z^{7}}{41287680}+\frac{% 2447z^{8}}{1321205760}+O(z^{9}).$

NTCIR12-MathWiki-17rate: 2

$1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4}.\!$

NTCIR12-MathWiki-17rate: 0

$\mu_{3,1}=\mu_{3,1}-r=\frac{14}{3}-5=\frac{-1}{3}$

NTCIR12-MathWiki-17rate: 0

$\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2^{5}% }{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{10% n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)=\pi\!$

NTCIR12-MathWiki-17rate: 1

$\sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{1}=\frac{1}{1}-\frac{% 1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\arctan{1}=\frac{\pi}{4}\!$

NTCIR12-MathWiki-17rate: 1

$\pi=<mtpl>{{1}}+\frac{{1}}{{2}}+\frac{{1}}{{3}}+\frac{{1}}{{4}}-\frac{{1}}{{5}% }+\frac{{1}}{{6}}+\frac{{1}}{{7}}+\frac{{1}}{{8}}+\frac{{1}}{{9}}-\frac{{1}}{{% 10}}+\frac{{1}}{{11}}+\frac{{1}}{{12}}-\frac{{1}}{{13}}+\cdots\!$

NTCIR12-MathWiki-17rate: 0

$\prod_{n=1}^{\infty}\left(\frac{4n^{2}}{4n^{2}-1}\right)=\frac{2}{1}\cdot\frac% {2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot% \frac{8}{7}\cdot\frac{8}{9}\cdots$

NTCIR12-MathWiki-17rate: 0

$\sum_{n=1}^{\infty}\frac{1}{n{2n\choose n}}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}% {5}+\frac{1}{7}-\frac{1}{8}+\cdots$

NTCIR12-MathWiki-17rate: 1

$\ln 2=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots.$

NTCIR12-MathWiki-17rate: 0

$x_{1}=-\tfrac{3}{4}$

NTCIR12-MathWiki-17rate: 0

$\alpha_{1}=\frac{1}{2}-m$

NTCIR12-MathWiki-17rate: 1

$\pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+% \frac{4}{13}-\cdots$

NTCIR12-MathWiki-17rate: 1

$=1-\tfrac{1}{6}=\tfrac{5}{6}$

NTCIR12-MathWiki-17rate: 0

$\frac{1}{4}\int_{0}^{1}\frac{x^{8}(1-x)^{8}}{1+x^{2}}\,dx=\pi-\frac{47\,171}{1% 5\,015}$

NTCIR12-MathWiki-17rate: 1

$H_{2n}-H_{n}\sim\log 2-\frac{1}{4n}+\frac{1}{16n^{2}}-\frac{1}{128n^{4}}+\frac% {1}{256n^{6}}-\frac{17}{4096n^{8}}+\cdots,$

NTCIR12-MathWiki-17rate: 3

$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-% \frac{1}{10}-\frac{1}{12}+\cdots$

NTCIR12-MathWiki-17rate: 1

$1+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+\cdots=\sum_{n=1}^{\infty}{1% \over n}.$

NTCIR12-MathWiki-17rate: 1

$\sum\limits_{n=1}^{\infty}{(-1)^{n+1}\over n}=1-{1\over 2}+{1\over 3}-{1\over 4% }+{1\over 5}-\cdots$

NTCIR12-MathWiki-17rate: 2

$1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4},\!$

NTCIR12-MathWiki-17rate: 0

$\operatorname{Li}_{2}\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_{2}% \left(\frac{1}{9}\right)=\frac{{\pi}^{2}}{18}-\frac{\ln^{2}3}{6}$

NTCIR12-MathWiki-17rate: 1

$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1807}+\cdots.$

NTCIR12-MathWiki-17rate: 0

$L_{6}\lambda=-\frac{2}{1+\lambda}+\frac{60}{2+\lambda}-\frac{420}{3+\lambda}+% \frac{1120}{4+\lambda}-\frac{1260}{5+\lambda}+\frac{504}{6+\lambda}.$

NTCIR12-MathWiki-17rate: 1

$i=1-(1-\frac{1}{2})\alpha=1-\frac{\alpha}{2}$

NTCIR12-MathWiki-17rate: 1

$\scriptstyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4}=0.7853% 981\ldots$

NTCIR12-MathWiki-18rate: 2

$\frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^{r}\,(1-p)^{k}\,$

NTCIR12-MathWiki-18rate: 3

$f(p)=\frac{(n+1)!}{k!(n-k)!}p^{k}(1-p)^{n-k}\,\text{ for }0\leq p\leq 1$

NTCIR12-MathWiki-18rate: 0

$P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}$

NTCIR12-MathWiki-18rate: 1

$=\sum{n!\over j_{1}!j_{2}!\cdots j_{n-k+1}!}\left({x_{1}\over 1!}\right)^{j_{1% }}\left({x_{2}\over 2!}\right)^{j_{2}}\cdots\left({x_{n-k+1}\over(n-k+1)!}% \right)^{j_{n-k+1}},$

NTCIR12-MathWiki-18rate: 1

$\theta_{n}(x)=\frac{n!}{(-2)^{n}}\,L_{n}^{-2n-1}(2x)$

NTCIR12-MathWiki-18rate: 1

${n\choose i}=\frac{n!}{i!(n-i)!}$

NTCIR12-MathWiki-18rate: 1

${n\choose i}=\frac{n!}{i!(n-i)!}$

NTCIR12-MathWiki-18rate: 0

$\operatorname{Var}[X]=np(1-p).$

NTCIR12-MathWiki-18rate: 1

${n\choose k}=\frac{n!}{k!(n-k)!}$

NTCIR12-MathWiki-18rate: 1

${n\choose k}=\frac{n!}{k!\,(n-k)!}$

NTCIR12-MathWiki-18rate: 0

$\alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)$

NTCIR12-MathWiki-18rate: 1

$w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.$

NTCIR12-MathWiki-18rate: 1

$D_{f}=\frac{L!}{N_{f}!(L-N_{f})!}.$

NTCIR12-MathWiki-18rate: 2

${\left({{N}\atop{N_{R}}}\right)}=\frac{N!}{N_{R}!(N-N_{R})!}$

NTCIR12-MathWiki-18rate: 1

$P_{N}(N_{R})=\frac{N!}{2^{N}N_{R}!(N-N_{R})!}$

NTCIR12-MathWiki-18rate: 1

${n\choose k}=\frac{n!}{k!\,(n-k)!},$

NTCIR12-MathWiki-18rate: 0

$f(x)-P_{n-1}(x)=\frac{f^{(n)}(\xi)}{n!}\prod_{i=1}^{n}(x-x_{i})$

NTCIR12-MathWiki-18rate: 3

$f(r|H=h,T=t)=\frac{(h+t+1)!}{h!\,\,t!}\;r^{h}\,(1-r)^{t}.\!$

NTCIR12-MathWiki-18rate: 1

${\left({{n}\atop{k}}\right)}=\frac{n!}{k!(n-k)!},$

NTCIR12-MathWiki-18rate: 1

$\Omega={\left(q+N^{\prime}-1\right)!\over q!(N^{\prime}-1)!}$

NTCIR12-MathWiki-18rate: 1

$E_{n}(x)=\frac{n!}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{n}}\,\mathrm{d}t=\frac{n!}{% \sqrt{\pi}}\sum_{p=0}^{\infty}(-1)^{p}\frac{x^{np+1}}{(np+1)p!}\,.$

NTCIR12-MathWiki-18rate: 0

$\frac{n!}{\prod_{i=1}^{k}x_{i}!}$

NTCIR12-MathWiki-18rate: 1

$w(n_{i},g_{i})=\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}\ .$

NTCIR12-MathWiki-18rate: 0

${\left({{p}\atop{n}}\right)}=\frac{p!}{n!(p-n)!}.$

NTCIR12-MathWiki-18rate: 1

${n\choose k}=\frac{n!}{k!(n-k)!}.$

NTCIR12-MathWiki-18rate: 1

$\int x^{k_{1}}\cdots x^{k_{2N}}\,\exp\left(-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x% _{i}x_{j}\right)\,d^{n}x=\sqrt{\frac{(2\pi)^{n}}{\det A}}\,\frac{1}{2^{N}N!}\,% \sum_{\sigma\in S_{2N}}(A^{-1})^{k_{\sigma(1)}k_{\sigma(2)}}\cdots(A^{-1})^{k_% {\sigma(2N-1)}k_{\sigma(2N)}}$

NTCIR12-MathWiki-18rate: 1

${n\choose k}={n!\over k!(n-k)!}$

NTCIR12-MathWiki-18rate: 1

$=\left(\ell_{A}\right)^{n_{A}}\left(\ell_{B}\right)^{n_{B}}\left(\underbrace{% \frac{N!}{n_{A}!n_{B}!}}_{combination}\right)\left(\underbrace{\frac{n_{A}\pi^% {n_{A}/2}}{(n_{A}/2)!}(2E_{A})^{\frac{n_{A}-1}{2}}}_{n_{A}-sphere}\right)\left% (\underbrace{\frac{n_{B}\pi^{n_{B}/2}}{(n_{B}/2)!}(2E_{B})^{\frac{n_{B}-1}{2}}% }_{n_{B}-sphere}\right)$

NTCIR12-MathWiki-18rate: 0

$\frac{n!}{l_{1}!l_{2}!\cdots l_{k}!}\prod_{i<j}(l_{i}-l_{j})$

NTCIR12-MathWiki-18rate: 0

${n\choose m}=\frac{n!}{m!\,(n-m)!}$

NTCIR12-MathWiki-18rate: 1

$|x_{1}x_{2}\cdots x_{N};S\rangle=\frac{\prod_{j}n_{j}!}{N!}\sum_{p}|x_{p(1)}% \rangle|x_{p(2)}\rangle\cdots|x_{p(N)}\rangle$

NTCIR12-MathWiki-18rate: 2

$P(k_{t}=k)={K\choose k}p_{t}^{k}(1-p_{t})^{K-k},\,\quad\quad k=0,1,\dots,K.$

NTCIR12-MathWiki-18rate: 0

$\mu(\sigma,\tau)=(-1)^{n-r}(2!)^{r_{3}}(3!)^{r_{4}}\cdots((n-1)!)^{r_{n}}$

NTCIR12-MathWiki-18rate: 1

${n\choose p}=\frac{n!}{p!(n-p)!}$

NTCIR12-MathWiki-18rate: 0

$\mathrm{Ran}\;e_{i}(T)=\mathrm{Ker}(T-\lambda_{i})^{\nu_{i}}.$

NTCIR12-MathWiki-18rate: 0

$2^{N}p^{K}(1-p)^{N-K}W\!.$

NTCIR12-MathWiki-18rate: 1

$\Omega_{conf}\,=\,\frac{N_{S}!}{N!(N_{S}-N)!}$

NTCIR12-MathWiki-18rate: 3

$\operatorname{Pr}(Y_{i}=y_{i}\mid\mathbf{X}_{i})={n_{i}\choose y_{i}}p_{i}^{y_% {i}}(1-p_{i})^{n_{i}-y_{i}}={n_{i}\choose y_{i}}\left(\frac{1}{1+e^{-symbol% \beta\cdot\mathbf{X}_{i}}}\right)^{y_{i}}\left(1-\frac{1}{1+e^{-symbol\beta% \cdot\mathbf{X}_{i}}}\right)^{n_{i}-y_{i}}$

NTCIR12-MathWiki-18rate: 1

${n\choose k}={n!\over k!(n-k)!}$

NTCIR12-MathWiki-18rate: 0

$W=N!\prod_{i}\frac{1}{N_{i}!}$

NTCIR12-MathWiki-18rate: 2

$f(k)={n\choose k}p^{k}(1-p)^{n-k}$

NTCIR12-MathWiki-18rate: 1

$\begin{aligned}\displaystyle W&\displaystyle=\frac{N!}{N_{a}!(N-N_{a})!}\times% \frac{(N-N_{a})!}{N_{b}!(N-N_{a}-N_{b})!}~{}\times\frac{(N-N_{a}-N_{b})!}{N_{c% }!(N-N_{a}-N_{b}-N_{c})!}\times\ldots\times\frac{(N-\ldots-N_{l})!}{N_{k}!(N-% \ldots-N_{l}-N_{k})!}=\\ \\ &\displaystyle=\frac{N!}{N_{a}!N_{b}!N_{c}!\ldots N_{k}!(N-\ldots-N_{l}-N_{k})% !}\end{aligned}$

NTCIR12-MathWiki-18rate: 0

$L^{*}=\prod_{n=1}^{N}{P_{i}^{Y_{i}}}\left(1-P_{i}\right)^{1-Y_{i}}$

NTCIR12-MathWiki-18rate: 0

$\frac{d^{\alpha_{i}}}{dx_{i}^{\alpha_{i}}}x_{i}^{\beta_{i}}=\frac{\beta_{i}!}{% (\beta_{i}-\alpha_{i})!}x_{i}^{\beta_{i}-\alpha_{i}}$

NTCIR12-MathWiki-18rate: 0

$\frac{n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}\frac{K!}{k_{m}!k_{m+1}!}=\frac{n!}{k_% {1}!k_{2}!\cdots k_{m+1}!}.$

NTCIR12-MathWiki-18rate: 0

${N\choose n_{1}}{N-n_{1}\choose n_{2}}{N-n_{1}-n_{2}\choose n_{3}}...=\frac{N!% }{(N-n_{1})!n_{1}!}\frac{(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}\frac{(N-n_{1}-n_{% 2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}....$

NTCIR12-MathWiki-18rate: 1

$W(n,N)={N\choose n}={{N!}\over{n!(N-n)!}}$

NTCIR12-MathWiki-18rate: 0

$R_{1}=\frac{1}{2!}f^{\prime\prime}(\xi_{n})(\alpha-x_{n})^{2}\,,$

NTCIR12-MathWiki-18rate: 3

${n!\over(k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\,du+O(du^{2}),$

NTCIR12-MathWiki-18rate: 4

$f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)$

NTCIR12-MathWiki-18rate: 0

$P-Q_{i}A_{i}=O((x-\lambda_{i})^{\nu_{i}})\qquad$

NTCIR12-MathWiki-18rate: 1

$Z=\frac{1}{N!h^{3N}}\int\,\exp[-\beta H(p_{1}\cdots p_{N},x_{1}\cdots x_{N})]% \;d^{3}p_{1}\cdots d^{3}p_{N}\,d^{3}x_{1}\cdots d^{3}x_{N}$

NTCIR12-MathWiki-18rate: 1

${\textstyle\left({{n}\atop{r}}\right)}=\tfrac{n!}{r!(n-r)!}$

NTCIR12-MathWiki-18rate: 0

$\alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)$

NTCIR12-MathWiki-18rate: 4

$p_{n}(k)=\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}.$

NTCIR12-MathWiki-18rate: 1

${n\choose r}={{n!}\over{r!(n-r)!}}={52\choose 5}={{52!}\over{5!(52-5)!}}=2,598% ,960$

NTCIR12-MathWiki-18rate: 1

$\frac{1}{p!}\Delta_{Q}^{p}f(p,q,r)=\frac{1}{(n-p)!}\Delta_{P}^{n-p}f(a,b,c).$

NTCIR12-MathWiki-18rate: 1

$\begin{array}[]{rcl}\frac{1}{N}\log W&=&\frac{1}{N}\log\frac{N!}{n_{1}!\,n_{2}% !\,\cdots\,n_{m}!}\\ \\ &=&\frac{1}{N}\log\frac{N!}{(Np_{1})!\,(Np_{2})!\,\cdots\,(Np_{m})!}\\ \\ &=&\frac{1}{N}\left(\log N!-\sum_{i=1}^{m}\log((Np_{i})!)\right).\end{array}$

NTCIR12-MathWiki-18rate: 1

$I_{B}=\frac{\log(N!)-\sum_{i=1}^{K}(\log(n_{i}!))}{N}$

NTCIR12-MathWiki-18rate: 0

$\alpha=4p(1-p).\!$

NTCIR12-MathWiki-18rate: 2

$C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}\frac{N!}{n!(N-n)!}q^{n}{(1-q)}^{N-n}{[S_{0}% {(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}$

NTCIR12-MathWiki-18rate: 0

$|C_{k}|\leq c^{\frac{k!}{n!(k-n)!}}$

NTCIR12-MathWiki-18rate: 3

$\frac{n!}{(n-r)!r!}\mu^{r}(1-\mu)^{n-r}\rightarrow e^{-\lambda}\frac{\lambda^{% r}}{r!}.$

NTCIR12-MathWiki-18rate: 3

$f(p)={(n+1)!\over s!(n-s)!}p^{s}(1-p)^{n-s}.$

NTCIR12-MathWiki-18rate: 2

$S_{k}={\left({{m}\atop{k}}\right)}\frac{(m-k)!}{m!}=\frac{1}{k!}.$

NTCIR12-MathWiki-18rate: 3

$L(\theta;A,B)=\frac{(A+B)!}{A!B!}\theta^{A}(1-\theta)^{B},$

NTCIR12-MathWiki-18rate: 0

$P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}$

NTCIR12-MathWiki-18rate: 4

$P_{i}^{x}=\frac{N!}{n_{x}!(N-n_{x})!}p_{x}^{n_{x}}(1-p_{x})^{N-n_{x}}$

NTCIR12-MathWiki-18rate: 0

$e^{-n\lambda}\lambda^{(x_{1}+x_{2}+\cdots+x_{n})}\cdot{1\over x_{1}!x_{2}!% \cdots x_{n}!}\,$

NTCIR12-MathWiki-18rate: 0

$s^{2}=np(1-p)$

NTCIR12-MathWiki-18rate: 0

$s^{2}=np(1-p)$

NTCIR12-MathWiki-19rate: 0

$\begin{aligned}\displaystyle\nabla\cdotsymbol{\tau}\equiv\cfrac{\partial s_{ij% }}{\partial x_{i}}&\displaystyle=\mu\left[\cfrac{\partial}{\partial x_{i}}% \left(\cfrac{\partial u_{i}}{\partial x_{j}}+\cfrac{\partial u_{j}}{\partial x% _{i}}\right)\right]+\lambda~{}\left[\cfrac{\partial}{\partial x_{i}}\left(% \cfrac{\partial u_{k}}{\partial x_{k}}\right)\right]\delta_{ij}\\ &\displaystyle=\mu~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}\partial x_{j}}+% \mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\lambda~{}\cfrac% {\partial^{2}u_{k}}{\partial x_{k}\partial x_{j}}\\ &\displaystyle=(\mu+\lambda)~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}% \partial x_{j}}+\mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}^{2}}\\ &\displaystyle\equiv(\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})+\mu~{}\nabla% ^{2}\mathbf{u}~{}.\end{aligned}$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=\frac{\partial% ^{2}T_{ij}}{\partial x_{i}\partial x_{j}},\quad(*)$

NTCIR12-MathWiki-19rate: 0

$\frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\nabla^{2}p=\rho_{0}% \frac{\partial^{2}\hat{T}_{ij}}{\partial x_{i}\partial x_{j}},\quad\,\text{% where}\quad\hat{T}_{ij}=v_{i}v_{j}.$

NTCIR12-MathWiki-19rate: 1

$\begin{aligned}\displaystyle\mathbb{S}&\displaystyle=\,\begin{pmatrix}S_{xx}&S% _{xy}\\ S_{yx}&S_{yy}\end{pmatrix}\,=\,\mathbb{I}\,\left(\frac{c_{g}}{c_{p}}-\frac{1}{% 2}\right)\,E\,+\,\frac{1}{k^{2}}\,\begin{pmatrix}k_{x}\,k_{x}&k_{x}\,k_{y}\\ k_{y}\,k_{x}&k_{y}\,k_{y}\end{pmatrix}\,\frac{c_{g}}{c_{p}}\,E,\\ \displaystyle\mathbb{I}&\displaystyle=\,\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\quad\,\text{and}\\ \displaystyle\nabla symbol{U}&\displaystyle=\,\begin{pmatrix}\displaystyle% \frac{\partial U_{x}}{\partial x}&\displaystyle\frac{\partial U_{y}}{\partial x% }\\ \displaystyle\frac{\partial U_{x}}{\partial y}&\displaystyle\frac{\partial U_{% y}}{\partial y}\end{pmatrix},\end{aligned}$

NTCIR12-MathWiki-19rate: 4

$H_{ij}=\begin{bmatrix}{\partial^{2}V_{ij}\over\partial x_{i}\partial x_{j}}&{% \partial^{2}V_{ij}\over\partial x_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over% \partial x_{i}\partial z_{j}}\\ {\partial^{2}V_{ij}\over\partial y_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial y_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial y_{i}\partial z% _{j}}\\ {\partial^{2}V_{ij}\over\partial z_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial z_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial z_{i}\partial z% _{j}}\end{bmatrix}$

NTCIR12-MathWiki-19rate: 0

$v_{x}{\partial c_{A}\over\partial x}+v_{y}{\partial c_{A}\over\partial y}=D_{% AB}{\partial^{2}c_{A}\over\partial y^{2}}$

NTCIR12-MathWiki-19rate: 1

$X=\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots&x_{1n}\\ x_{12}&x_{22}&x_{23}&\cdots&x_{2n}\\ x_{13}&x_{23}&x_{33}&\cdots&x_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1n}&x_{2n}&x_{3n}&\cdots&x_{nn}\end{vmatrix},D=\begin{vmatrix}2\frac{% \partial}{\partial x_{11}}&\frac{\partial}{\partial x_{12}}&\frac{\partial}{% \partial x_{13}}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \frac{\partial}{\partial x_{12}}&2\frac{\partial}{\partial x_{22}}&\frac{% \partial}{\partial x_{23}}&\cdots&\frac{\partial}{\partial x_{2n}}\\ \frac{\partial}{\partial x_{13}}&\frac{\partial}{\partial x_{23}}&2\frac{% \partial}{\partial x_{33}}&\cdots&\frac{\partial}{\partial x_{3n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{1n}}&\frac{\partial}{\partial x_{2n}}&\frac{% \partial}{\partial x_{3n}}&\cdots&2\frac{\partial}{\partial x_{nn}}\end{vmatrix}$

NTCIR12-MathWiki-19rate: 0

$\begin{aligned}\displaystyle\sigma_{ij}&\displaystyle=-\begin{pmatrix}p&0&0\\ 0&p&0\\ 0&0&p\end{pmatrix}+\mu\begin{pmatrix}2\frac{\partial u}{\partial x}&\frac{% \partial u}{\partial y}+\frac{\partial v}{\partial x}&\frac{\partial u}{% \partial z}+\frac{\partial w}{\partial x}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}&2\frac{\partial v}% {\partial y}&\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\\ \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}&\frac{\partial w}{% \partial y}+\frac{\partial v}{\partial z}&2\frac{\partial w}{\partial z}\end{% pmatrix}\\ &\displaystyle=-pI+\mu(\nabla\mathbf{v}+(\nabla\mathbf{v})^{T})\end{aligned}$

NTCIR12-MathWiki-19rate: 1

$\begin{pmatrix}\bar{x}_{1}\\ \bar{x}_{2}\\ \bar{x}_{3}\end{pmatrix}=\begin{pmatrix}\frac{\partial\bar{x}_{1}}{\partial x_% {1}}&\frac{\partial\bar{x}_{1}}{\partial x_{2}}&\frac{\partial\bar{x}_{1}}{% \partial x_{3}}\\ \frac{\partial\bar{x}_{2}}{\partial x_{1}}&\frac{\partial\bar{x}_{2}}{\partial x% _{2}}&\frac{\partial\bar{x}_{2}}{\partial x_{3}}\\ \frac{\partial\bar{x}_{3}}{\partial x_{1}}&\frac{\partial\bar{x}_{3}}{\partial x% _{2}}&\frac{\partial\bar{x}_{3}}{\partial x_{3}}\end{pmatrix}\begin{pmatrix}x_% {1}\\ x_{2}\\ x_{3}\end{pmatrix}$

NTCIR12-MathWiki-19rate: 1

$Df(x,y)=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{% \partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}$

NTCIR12-MathWiki-19rate: 1

$\Omega=\begin{vmatrix}\frac{\partial}{\partial x_{11}}&\cdots&\frac{\partial}{% \partial x_{1n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{n1}}&\cdots&\frac{\partial}{\partial x_{nn}}\end{% vmatrix}.$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=\sum_{k}\left(\frac{% \partial y}{\partial u_{k}}\frac{\partial^{2}u_{k}}{\partial x_{i}\partial x_{% j}}\right)+\sum_{k,\ell}\left(\frac{\partial^{2}y}{\partial u_{k}\partial u_{% \ell}}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{\ell}}{\partial x% _{j}}\right).$

NTCIR12-MathWiki-19rate: 0

$\mathbf{J}=\left(\begin{array}[]{cccc}\frac{\partial\sigma_{1}}{\partial% \lambda_{1}}&\frac{\partial\sigma_{1}}{\partial\lambda_{2}}&\dots&\frac{% \partial\sigma_{1}}{\partial\lambda_{n}}\\ \frac{\partial\sigma_{2}}{\partial\lambda_{1}}&\frac{\partial\sigma_{2}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{2}}{\partial\lambda_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\sigma_{n}}{\partial\lambda_{1}}&\frac{\partial\sigma_{n}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{n}}{\partial\lambda_{n}}\end{% array}\right).$

NTCIR12-MathWiki-19rate: 1

$\mathbf{J}=\begin{bmatrix}\cfrac{\partial x_{1}}{\partial q^{1}}&\cfrac{% \partial x_{1}}{\partial q^{2}}&\cfrac{\partial x_{1}}{\partial q^{3}}\\ \cfrac{\partial x_{2}}{\partial q^{1}}&\cfrac{\partial x_{2}}{\partial q^{2}}&% \cfrac{\partial x_{2}}{\partial q^{3}}\\ \cfrac{\partial x_{3}}{\partial q^{1}}&\cfrac{\partial x_{3}}{\partial q^{2}}&% \cfrac{\partial x_{3}}{\partial q^{3}}\\ \end{bmatrix},\quad\mathbf{J}^{-1}=\begin{bmatrix}\cfrac{\partial q^{1}}{% \partial x_{1}}&\cfrac{\partial q^{1}}{\partial x_{2}}&\cfrac{\partial q^{1}}{% \partial x_{3}}\\ \cfrac{\partial q^{2}}{\partial x_{1}}&\cfrac{\partial q^{2}}{\partial x_{2}}&% \cfrac{\partial q^{2}}{\partial x_{3}}\\ \cfrac{\partial q^{3}}{\partial x_{1}}&\cfrac{\partial q^{3}}{\partial x_{2}}&% \cfrac{\partial q^{3}}{\partial x_{3}}\\ \end{bmatrix}$

NTCIR12-MathWiki-19rate: 0

$\iiint_{V}\left[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{% \partial x_{j}}\right]dV=\iiint_{V}\left[-\frac{\partial P}{\partial x_{i}}+% \nu\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}+f_{i}\right]dV$

NTCIR12-MathWiki-19rate: 1

$\mathbf{\varepsilon}=\begin{bmatrix}\dfrac{\partial v_{1}}{\partial S_{1}}&% \cdots&\dfrac{\partial v_{1}}{\partial S_{m}}\\ \vdots&\ddots&\vdots\\ \dfrac{\partial v_{n}}{\partial S_{1}}&\cdots&\dfrac{\partial v_{n}}{\partial S% _{m}}\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$V_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial\Sigma}{\partial\theta_{m}}=\begin{bmatrix}\frac{\partial\Sigma_% {1,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{1,2}}{\partial\theta_{m}}&% \cdots&\frac{\partial\Sigma_{1,N}}{\partial\theta_{m}}\\ \\ \frac{\partial\Sigma_{2,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{2,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{2,N}}{\partial\theta_{m}}\\ \\ \vdots&\vdots&\ddots&\vdots\\ \\ \frac{\partial\Sigma_{N,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{N,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{N,N}}{\partial\theta_{m}}\end% {bmatrix}.$

NTCIR12-MathWiki-19rate: 0

$\tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}% {\partial x_{i}}-\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}\right)+\kappa% \delta_{ij}\nabla\cdot\mathbf{v}$

NTCIR12-MathWiki-19rate: 1

$\begin{bmatrix}\dfrac{\partial x_{1}}{\partial x}&\dfrac{\partial x_{1}}{% \partial y}&\dfrac{\partial x_{1}}{\partial z}\\ \dfrac{\partial x_{2}}{\partial x}&\dfrac{\partial x_{2}}{\partial y}&\dfrac{% \partial x_{2}}{\partial z}\\ \dfrac{\partial x_{3}}{\partial x}&\dfrac{\partial x_{3}}{\partial y}&\dfrac{% \partial x_{3}}{\partial z}\end{bmatrix}$

NTCIR12-MathWiki-19rate: 1

$E_{ij}:=\frac{1}{2}\left[\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}+\frac{\partial u_{k}}{\partial x_{i}}\,\frac{\partial u_% {k}}{\partial x_{j}}\right]\,.$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial^{2}S}{\partial\beta_{j}\partial\beta_{k}}$

NTCIR12-MathWiki-19rate: 0

$A_{ij}=\frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial% \alpha_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial^{2}\psi}{\partial% \theta_{i}\partial\theta_{j}}$

NTCIR12-MathWiki-19rate: 2

$H(f,g)=\begin{bmatrix}0&\dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial g}{% \partial x_{2}}&\cdots&\dfrac{\partial g}{\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&% \dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{% 2}f}{\partial x_{1}\,\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{2}}&\dfrac{\partial^{2}f}{\partial x_{2}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\dfrac{% \partial^{2}f}{\partial x_{2}\,\partial x_{n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial g}{\partial x_{n}}&\dfrac{\partial^{2}f}{\partial x_{n}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{2}}&\cdots&% \dfrac{\partial^{2}f}{\partial x_{n}^{2}}\end{bmatrix}$

NTCIR12-MathWiki-19rate: 2

$H=\begin{bmatrix}\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&\dfrac{\partial^{2}% f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{1}% \,\partial x_{n}}\\ \dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{2}^{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{n% }}\\ \vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{n}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{n}^{2% }}\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 0

$Lu=a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+b_{i}(x)\frac{% \partial u}{\partial x_{i}}+c(x)u,\qquad x\in\Omega.$

NTCIR12-MathWiki-19rate: 1

$q_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.$

NTCIR12-MathWiki-19rate: 0

$\frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!$

NTCIR12-MathWiki-19rate: 1

$\left[\begin{matrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\frac{\partial u_{x}}{\partial x}&% \frac{1}{2}\left(\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{% \partial x}\right)&\frac{1}{2}\left(\frac{\partial u_{x}}{\partial z}+\frac{% \partial u_{z}}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{% \partial y}\right)&\frac{\partial u_{y}}{\partial y}&\frac{1}{2}\left(\frac{% \partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{% \partial z}\right)&\frac{1}{2}\left(\frac{\partial u_{z}}{\partial y}+\frac{% \partial u_{y}}{\partial z}\right)&\frac{\partial u_{z}}{\partial z}\\ \end{matrix}\right]\,\!$

NTCIR12-MathWiki-19rate: 2

$H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}\\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 0

$\dot{y}=\frac{\partial y}{\partial x_{j}}\dot{x}_{j}+\frac{1}{2}\frac{\partial% ^{2}y}{\partial x_{k}\,\partial x_{l}}g_{km}g_{ml}.$

NTCIR12-MathWiki-19rate: 1

$J_{F}=\left|\begin{matrix}\frac{\partial f_{1}}{\partial X_{1}}&\cdots&\frac{% \partial f_{1}}{\partial X_{N}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{N}}{\partial X_{1}}&\cdots&\frac{\partial f_{N}}{\partial X_% {N}}\end{matrix}\right|,$

NTCIR12-MathWiki-19rate: 1

$\mathbf{J}_{\mathbf{F}}(r,\theta,\varphi)=\begin{bmatrix}\dfrac{\partial x}{% \partial r}&\dfrac{\partial x}{\partial\theta}&\dfrac{\partial x}{\partial% \varphi}\\ \dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\theta}&\dfrac{% \partial y}{\partial\varphi}\\ \dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial\theta}&\dfrac{% \partial z}{\partial\varphi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\varphi% &r\cos\theta\cos\varphi&-r\sin\theta\sin\varphi\\ \sin\theta\sin\varphi&r\cos\theta\sin\varphi&r\sin\theta\cos\varphi\\ \cos\theta&-r\sin\theta&0\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$\mathbf{J}_{\mathbf{F}}(x_{1},x_{2},x_{3})=\begin{bmatrix}\dfrac{\partial y_{1% }}{\partial x_{1}}&\dfrac{\partial y_{1}}{\partial x_{2}}&\dfrac{\partial y_{1% }}{\partial x_{3}}\\ \dfrac{\partial y_{2}}{\partial x_{1}}&\dfrac{\partial y_{2}}{\partial x_{2}}&% \dfrac{\partial y_{2}}{\partial x_{3}}\\ \dfrac{\partial y_{3}}{\partial x_{1}}&\dfrac{\partial y_{3}}{\partial x_{2}}&% \dfrac{\partial y_{3}}{\partial x_{3}}\\ \dfrac{\partial y_{4}}{\partial x_{1}}&\dfrac{\partial y_{4}}{\partial x_{2}}&% \dfrac{\partial y_{4}}{\partial x_{3}}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&0&5\\ 0&8x_{2}&-2\\ x_{3}\cos x_{1}&0&\sin x_{1}\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial^{2}u}{\partial x_{i}\partial y_{j}}-\frac{\partial^{2}u}{% \partial y_{i}\partial x_{j}}=0$

NTCIR12-MathWiki-19rate: 1

$\sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\sigma_{jk}\cfrac{% \partial^{2}u_{j}}{\partial x_{1}\partial x_{k}}$

NTCIR12-MathWiki-19rate: 1

$\delta=\frac{1}{4\pi}\iint_{R_{st}}\frac{\begin{vmatrix}x&y&z\\ \dfrac{\partial x}{\partial s}&\dfrac{\partial y}{\partial s}&\dfrac{\partial z% }{\partial s}\\ \dfrac{\partial x}{\partial t}&\dfrac{\partial y}{\partial t}&\dfrac{\partial z% }{\partial t}\end{vmatrix}}{(x^{2}+y^{2}+z^{2})\sqrt{x^{2}+y^{2}+z^{2}}}dsdt.$

NTCIR12-MathWiki-19rate: 0

$\frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{% \partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu\frac% {\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{\partial\tau_{ij% }}{\partial x_{j}}.$

NTCIR12-MathWiki-19rate: 0

$\frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!$

NTCIR12-MathWiki-19rate: 1

$\mathbf{A}=\begin{bmatrix}\frac{\partial F_{1}(\vec{q})}{\partial p_{1}}&\frac% {\partial F_{1}(\vec{q})}{\partial p_{2}}&\cdots&\frac{\partial F_{1}(\vec{q})% }{\partial p_{n}}\\ \frac{\partial F_{2}(\vec{q})}{\partial p_{1}}&\cdots&\frac{\partial F_{2}(% \vec{q})}{\partial p_{n-1}}&\frac{\partial F_{2}(\vec{q})}{\partial p_{n}}\\ \vdots&\frac{\partial F_{j}(\vec{q})}{\partial p_{i}}&\vdots&\vdots\\ \frac{\partial F_{m}(\vec{q})}{\partial p_{1}}&\frac{\partial F_{m}(\vec{q})}{% \partial p_{2}}&\cdots&\frac{\partial F_{m}(\vec{q})}{\partial p_{n}}\\ \end{bmatrix}\!$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial\mathbf{\phi}(\mathbf{X})}{\partial\mathbf{X}}=\begin{bmatrix}% \frac{\partial\mathbf{\phi}}{\partial x_{1,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{1,q}}\\ \vdots&\ddots&\vdots\\ \frac{\partial\mathbf{\phi}}{\partial x_{n,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{n,q}}\\ \end{bmatrix}$

NTCIR12-MathWiki-19rate: 1

$\frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{11}}&\frac{\partial y}{\partial x_{12}}&\cdots&\frac{\partial y}{% \partial x_{1q}}\\ \frac{\partial y}{\partial x_{21}}&\frac{\partial y}{\partial x_{22}}&\cdots&% \frac{\partial y}{\partial x_{2q}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{p1}}&\frac{\partial y}{\partial x_{p2}}&\cdots&% \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$\left(\frac{\partial^{2}S}{\partial y\partial x}\right)=\left(\frac{\partial^{% 2}S}{\partial x\partial y}\right):\left(\frac{\partial^{2}V}{\partial y% \partial x}\right)=\left(\frac{\partial^{2}V}{\partial x\partial y}\right)$

NTCIR12-MathWiki-19rate: 2

$\textstyle\mbox{Hess}~{}(F)=\begin{pmatrix}\frac{\partial^{2}F}{\partial x^{2}% }&\frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial x% \partial z}\\ \frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial y^{2}% }&\frac{\partial^{2}F}{\partial y\partial z}\\ \frac{\partial^{2}F}{\partial x\partial z}&\frac{\partial^{2}F}{\partial y% \partial z}&\frac{\partial^{2}F}{\partial z^{2}}\end{pmatrix}.$

NTCIR12-MathWiki-19rate: 1

$\left.\frac{\partial^{2}S(z)}{\partial z_{i}\partial z_{j}}\right|_{z=0}=2h_{% ij}(0);$

NTCIR12-MathWiki-19rate: 1

$J=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{% \partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$D\varphi=\begin{bmatrix}\frac{\partial\varphi^{1}}{\partial x^{1}}&\frac{% \partial\varphi^{1}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{1}}{\partial x% ^{n}}\\ \frac{\partial\varphi^{2}}{\partial x^{1}}&\frac{\partial\varphi^{2}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{2}}{\partial x^{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\varphi^{m}}{\partial x^{1}}&\frac{\partial\varphi^{m}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{m}}{\partial x^{n}}\\ \end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$=-\left.\left(\begin{array}[]{cccc}\tfrac{\partial^{2}}{\partial\theta_{1}^{2}% }&\tfrac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}&\cdots&\tfrac{% \partial^{2}}{\partial\theta_{1}\partial\theta_{n}}\\ \tfrac{\partial^{2}}{\partial\theta_{2}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{2}^{2}}&\cdots&\tfrac{\partial^{2}}{\partial\theta_{2}% \partial\theta_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \tfrac{\partial^{2}}{\partial\theta_{n}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{n}\partial\theta_{2}}&\cdots&\tfrac{\partial^{2}}{\partial% \theta_{n}^{2}}\\ \end{array}\right)\ell(\theta)\right|_{\theta=\theta^{*}}$

NTCIR12-MathWiki-19rate: 0

$\frac{\partial^{2}f}{\partial y\,\partial x}=\frac{\partial}{\partial y}\left(% \frac{\partial f}{\partial x}\right)=(f_{x})_{y}=f_{xy}=\partial_{yx}f.$

NTCIR12-MathWiki-19rate: 0

$\begin{array}[]{lll}\displaystyle\frac{\partial f}{\partial x}(0,r_{o})=0,&% \displaystyle\frac{\partial^{2}f}{\partial x^{2}}(0,r_{o})=0,&\displaystyle% \frac{\partial^{3}f}{\partial x^{3}}(0,r_{o})\neq 0,\\ \displaystyle\frac{\partial f}{\partial r}(0,r_{o})=0,&\displaystyle\frac{% \partial^{2}f}{\partial r\partial x}(0,r_{o})\neq 0.\end{array}$

NTCIR12-MathWiki-19rate: 1

$\lambda_{ij}=\frac{\partial^{2}f}{\partial z_{i}\partial\bar{z}_{j}}$

NTCIR12-MathWiki-19rate: 2

$H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}&\frac{\partial^{2}f}{\partial x\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}&\frac{\partial^{2}f}{\partial y\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial z\,\partial x}&\frac{\partial^{2}f}{\partial z\,% \partial y}&\frac{\partial^{2}f}{\partial z^{2}}\end{bmatrix},$

NTCIR12-MathWiki-19rate: 1

$J=\begin{bmatrix}\dfrac{\partial\Delta P}{\partial\theta}&\dfrac{\partial% \Delta P}{\partial|V|}\\ \dfrac{\partial\Delta Q}{\partial\theta}&\dfrac{\partial\Delta Q}{\partial|V|}% \end{bmatrix}$

NTCIR12-MathWiki-19rate: 0

$\rho\frac{Du_{i}}{Dt}=-\frac{\partial p}{\partial x_{i}}+\mu\left(\frac{% \partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}\right),$

NTCIR12-MathWiki-19rate: 0

$W_{ijkl}=\frac{\partial^{2}F_{ij}}{\partial x_{k}\partial x_{l}}+\frac{% \partial^{2}F_{kl}}{\partial x_{i}\partial x_{j}}-\frac{\partial^{2}F_{il}}{% \partial x_{j}\partial x_{k}}-\frac{\partial^{2}F_{jk}}{\partial x_{i}\partial x% _{l}}$

NTCIR12-MathWiki-19rate: 0

$\begin{pmatrix}\frac{1}{c}\frac{\partial\phi}{\partial t^{\prime}}&\frac{% \partial\phi}{\partial x^{\prime}}&\frac{\partial\phi}{\partial y^{\prime}}&% \frac{\partial\phi}{\partial z^{\prime}}\end{pmatrix}=\begin{pmatrix}\frac{1}{% c}\frac{\partial\phi}{\partial t}&\frac{\partial\phi}{\partial x}&\frac{% \partial\phi}{\partial y}&\frac{\partial\phi}{\partial z}\end{pmatrix}\begin{% pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\,.$

NTCIR12-MathWiki-19rate: 1

$L=\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}+\sum_{i,j=1}^{n}a_{ij}% (x)\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}},$

NTCIR12-MathWiki-19rate: 2

$\frac{\partial^{2}\Pi}{\partial x_{i}\partial x_{j}}$

NTCIR12-MathWiki-19rate: 2

$\begin{bmatrix}\dfrac{dx_{1}^{*}}{dp_{1}}\\ \dfrac{dx_{2}^{*}}{dp_{1}}\\ \dfrac{dy_{2}^{*}}{dp_{1}}\end{bmatrix}=\begin{bmatrix}\dfrac{\partial^{2}U_{x% }}{\partial x_{1}\partial x_{1}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}% \partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial y_{2}}\\ \dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{2}}&\dfrac{\partial^{2}U_{% x}}{\partial x_{2}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial y_{2}% \partial x_{2}}\\ \dfrac{\partial^{2}U_{y}}{\partial x_{1}\partial y_{2}}&\dfrac{\partial^{2}U_{% y}}{\partial x_{2}\partial y_{2}}&\dfrac{\partial^{2}U_{y}}{\partial y_{2}% \partial y_{2}}\end{bmatrix}^{-1}\begin{bmatrix}-\dfrac{\partial^{2}U_{x}}{% \partial p_{1}\partial x_{1}}\\ -\dfrac{\partial^{2}U_{x}}{\partial p_{1}\partial x_{2}}\\ -\dfrac{\partial^{2}U_{y}}{\partial p_{1}\partial y_{2}}\end{bmatrix}$

NTCIR12-MathWiki-19rate: 0

$\sigma_{x}=\frac{\partial^{2}\Phi_{yy}}{\partial z\partial z}+\frac{\partial^{% 2}\Phi_{zz}}{\partial y\partial y}-2\frac{\partial^{2}\Phi_{yz}}{\partial y% \partial z}$

NTCIR12-MathWiki-19rate: 1

$\begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\ |U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\ |U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}=\begin{bmatrix% }\frac{2}{3}&\frac{1}{3}&0\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\end{bmatrix}.$

NTCIR12-MathWiki-19rate: 1

$U=\begin{bmatrix}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\ \frac{\omega}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\bar{\omega}}{\sqrt{3}}\\ \frac{\bar{\omega}}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\omega}{\sqrt{3}}\end{% bmatrix}\Rightarrow(|U_{i\alpha}|^{2})=\begin{bmatrix}\frac{1}{3}&\frac{1}{3}&% \frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$

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NTCIR12-MathWiki-20rate: 1

$S_{\sigma^{2}}=\sum_{i=1}^{m}(X_{i}-\overline{X})^{2},\,\text{ where }% \overline{X}=\frac{S_{\mu}}{m}$

NTCIR12-MathWiki-20rate: 1

$\mathrm{variance}=s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{% n-1}\!$

NTCIR12-MathWiki-20rate: 0

$\textstyle C_{n}=\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})(y_{i}-\bar{y}_{n})$

NTCIR12-MathWiki-20rate: 3

$r(\alpha,\beta)=\frac{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})(k^{\beta}_{i}-% \bar{k^{\beta}})}{\sqrt{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})^{2}}\sqrt{% \sum_{i}(k^{\beta}_{i}-\bar{k^{\beta}})^{2}}}.$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle\operatorname{E}(s^{2})&\displaystyle=% \operatorname{E}\left(\sum_{i=1}^{n}\frac{(x_{i}-\overline{x})^{2}}{n-1}\right% )\\ &\displaystyle=\frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^{n}\left[x_{i}-% \mu-\left(\overline{x}-\mu\right)\right]^{2}\right)\\ &\displaystyle=\frac{1}{n-1}\left[\sum_{i=1}^{n}\operatorname{Var}\left(x_{i}% \right)-n\operatorname{Var}\left(\overline{x}\right)\right]\end{aligned}$

NTCIR12-MathWiki-20rate: 1

$\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$

NTCIR12-MathWiki-20rate: 1

$s_{x}=\frac{s_{y}}{|m|}\sqrt{\frac{1}{n}+\frac{1}{k}+\frac{(y_{unk}-\bar{y})^{% 2}}{m^{2}\sum{(x_{i}-\bar{x})^{2}}}}$

NTCIR12-MathWiki-20rate: 0

$\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1}$

NTCIR12-MathWiki-20rate: 3

$\operatorname{simil}(x,y)=\frac{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})% (r_{y,i}-\bar{r_{y}})}{\sqrt{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})^{2% }\sum\limits_{i\in I_{xy}}(r_{y,i}-\bar{r_{y}})^{2}}}$

NTCIR12-MathWiki-20rate: 0

$s_{xy}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\bar{x})(y_{n}-\bar{y}).$

NTCIR12-MathWiki-20rate: 1

$\mathbf{C}=\sum_{i=1}^{n}(\mathbf{x_{i}}-\mathbf{\bar{x}})(\mathbf{x_{i}}-% \mathbf{\bar{x}})^{T}$

NTCIR12-MathWiki-20rate: 4

$r_{xy}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{(n-1)s_{x}s_% {y}}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum% \limits_{i=1}^{n}(x_{i}-\bar{x})^{2}\sum\limits_{i=1}^{n}(y_{i}-\bar{y})^{2}}},$

NTCIR12-MathWiki-20rate: 3

$\,\text{similarity}=\cos(\theta)={A\cdot B\over\|A\|\|B\|}=\frac{\sum\limits_{% i=1}^{n}{A_{i}\times B_{i}}}{\sqrt{\sum\limits_{i=1}^{n}{(A_{i})^{2}}}\times% \sqrt{\sum\limits_{i=1}^{n}{(B_{i})^{2}}}}$

NTCIR12-MathWiki-20rate: 2

$\displaystyle g_{1}=\frac{m_{3}}{m_{2}^{3/2}}=\frac{\frac{1}{n}\sum_{i=1}^{n}% \left(x_{i}-\bar{x}\right)^{3}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-% \bar{x}\right)^{2}\right)^{3/2}}\ ,$

NTCIR12-MathWiki-20rate: 1

$\frac{\sqrt{n}(\bar{X}-\mu_{0})}{\sqrt{\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2% }/(n-1)}}$

NTCIR12-MathWiki-20rate: 1

$\,\text{SSE}=\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}+\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{% 2}+\sum_{i=1}^{n}(Z_{i}-\bar{Z})^{2}$

NTCIR12-MathWiki-20rate: 3

$\frac{\sum_{k=1}^{n}(x_{k}-\bar{x})(y_{k}-\bar{y})}{\sqrt{\sum_{k=1}^{n}(x_{k}% -\bar{x})^{2}\sum_{k=1}^{n}(y_{k}-\bar{y})^{2}}}$

NTCIR12-MathWiki-20rate: 1

$\hat{\beta}=\frac{\tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})(y_{t}-\bar{y})}{% \tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})^{2}}\,,$

NTCIR12-MathWiki-20rate: 1

$\mathcal{L}(\overline{x},\Sigma)\propto\det(\Sigma)^{-n/2}\exp\left(-{1\over 2% }\sum_{i=1}^{n}(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(x_{i}-\overline{x}% )\right),$

NTCIR12-MathWiki-20rate: 0

${S\over n}={1\over n}\sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{% \mathrm{T}}$

NTCIR12-MathWiki-20rate: 0

$\sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)^{\mathrm{T}}=\sum_{i=1}^{n}(x_{i}-\bar{x}% )(x_{i}-\bar{x})^{\mathrm{T}}=S$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle\sum_{i=1}^{n}2(\hat{y}_{i}-\bar{y})(y_{i}-\hat{y}% _{i})&\displaystyle=\sum_{i=1}^{n}2\hat{b}\left((y_{i}-\bar{y})(x_{i}-\bar{x})% -\hat{b}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\left(\sum_{i=1}^{n}(y_{i}-\bar{y})(x_{i}-\bar{x})-\hat% {b}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\sum_{i=1}^{n}\left((y_{i}-\bar{y})(x_{i}-\bar{x})-(y_{% i}-\bar{y})(x_{i}-\bar{x})\right)\\ &\displaystyle=2\hat{b}\cdot 0=0.\end{aligned}$

NTCIR12-MathWiki-20rate: 1

$\hat{b}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{n}(x_{% i}-\bar{x})^{2}},$

NTCIR12-MathWiki-20rate: 1

$S_{X}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}\,% \text{ and }S_{Y}^{2}=\frac{1}{m-1}\sum_{i=1}^{m}\left(Y_{i}-\overline{Y}% \right)^{2}$

NTCIR12-MathWiki-20rate: 3

$r=\frac{\sum^{N}_{i=1}(X_{i}-\bar{X})(Y_{i}-\bar{Y})}{\sqrt{\sum^{N}_{i=1}(X_{% i}-\bar{X})^{2}}\sqrt{\sum^{N}_{i=1}(Y_{i}-\bar{Y})^{2}}}$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle SS_{\rm err}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}% -\widehat{y_{i}})^{2}\\ \displaystyle SS_{\rm tot}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}-\bar{y})^{2}\\ \displaystyle SS_{\rm reg}&\displaystyle=\sum_{i=1}^{N}\;(\widehat{y_{i}}-\bar% {y})^{2}\,\text{ and}\\ \displaystyle\bar{y}&\displaystyle=\frac{1}{N}\sum{}_{i=1}^{N}\;y_{i}.\end{aligned}$

NTCIR12-MathWiki-20rate: 1

$g^{\prime}(\beta)=-\frac{\psi(1/\beta)}{\beta^{2}}-\frac{\psi^{\prime}(1/\beta% )}{\beta^{3}}+\frac{1}{\beta^{2}}-\frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}(\log% |x_{i}-\mu|)^{2}}{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}}+\frac{(\sum_{i=1}^{N}|x_{% i}-\mu|^{\beta}\log|x_{i}-\mu|)^{2}}{(\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})^{2}}+% \frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}\log|x_{i}-\mu|}{\beta\sum_{i=1}^{N}|x_% {i}-\mu|^{\beta}}-\frac{\log(\frac{\beta}{N}\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})% }{\beta^{2}},$

NTCIR12-MathWiki-20rate: 1

$q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-% \bar{x},y-\bar{y})I(x,y)}$

NTCIR12-MathWiki-20rate: 1

$q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})I(x,y)}{\sum I(x,y)}$

NTCIR12-MathWiki-20rate: 0

$\textstyle v=\frac{1}{k}\sum_{i=1}^{k}(m_{i}-\bar{m})^{2}$

NTCIR12-MathWiki-20rate: 0

${\mathbf{W}}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\overline{\mathbf{x}})% (\mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}$

NTCIR12-MathWiki-20rate: 1

${\mathbf{W}}=\frac{\sum_{i=1}^{n_{x}}(\mathbf{x}_{i}-\overline{\mathbf{x}})(% \mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}+\sum_{i=1}^{n_{y}}(\mathbf{y}_{% i}-\overline{\mathbf{y}})(\mathbf{y}_{i}-\overline{\mathbf{y}})^{\prime}}{n_{x% }+n_{y}-2}$

NTCIR12-MathWiki-20rate: 1

$C=\frac{\hat{\mu}_{4}}{\hat{\sigma}^{4}}=\frac{\frac{1}{n}\sum_{i=1}^{n}(x_{i}% -\bar{x})^{4}}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}},$

NTCIR12-MathWiki-20rate: 0

$S=\sum_{i=1}^{n}(R_{i}-\bar{R})^{2},$

NTCIR12-MathWiki-20rate: 0

$\sigma^{2}_{X}=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\,{}}{n}.$

NTCIR12-MathWiki-20rate: 1

$g_{2}=\frac{m_{4}}{m_{2}^{2}}-3=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-% \overline{x})^{4}}{\left(\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}% \right)^{2}}-3$

NTCIR12-MathWiki-20rate: 1

$=\frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}% }{\left(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}}-3\,\frac{(n-1)^{2}}{(n-2% )\,(n-3)}$

NTCIR12-MathWiki-20rate: 0

$\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1},\quad$

NTCIR12-MathWiki-20rate: 1

$RMD(S)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\sum_{i=1}^{n}y_{% i}}$

NTCIR12-MathWiki-20rate: 0

$\sigma^{2}=\frac{\sum_{i=1}^{N}w_{i}(x_{i}-\overline{x}^{\,*})^{2}}{\sum_{i=1}% ^{N}w_{i}}$

NTCIR12-MathWiki-20rate: 2

$I=\frac{N}{\sum_{i}\sum_{j}w_{ij}}\frac{\sum_{i}\sum_{j}w_{ij}(X_{i}-\bar{X})(% X_{j}-\bar{X})}{\sum_{i}(X_{i}-\bar{X})^{2}}$

NTCIR12-MathWiki-20rate: 1

$\displaystyle\widehat{symbol\Sigma}={1\over n}\sum_{j=1}^{n}\left(\mathbf{x}_{% j}-\bar{\mathbf{x}}\right)\left(\mathbf{x}_{j}-\bar{\mathbf{x}}\right)^{T}$

NTCIR12-MathWiki-20rate: 1

$F=\frac{{\displaystyle\sum_{i=1}^{n}\left(\widehat{y_{i}}-\bar{y}\right)^{2}}/% {k}}{{\displaystyle{\sum_{j=1}^{k}}{\sum_{i=1}^{n_{j}}}\left(y_{ij}-\widehat{y% _{i}}\right)^{2}}/{(n-k-1)}}$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}&% \displaystyle=\sum_{i=1}^{n}(y_{i}-\overline{y}+\hat{y}_{i}-\hat{y}_{i})^{2}=% \sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})+\underbrace{(y_{i}-\hat{y}_{i})}_{\hat{% \varepsilon}_{i}})^{2}\\ &\displaystyle=\sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})^{2}+2\hat{\varepsilon}_{i}% (\hat{y}_{i}-\bar{y})+\hat{\varepsilon}_{i}^{2})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{y}_{i}-\bar{y})% \\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{\beta}_{0}+\hat% {\beta}_{1}x_{i1}+\cdots+\hat{\beta}_{p}x_{ip}-\overline{y})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2(\hat{\beta}_{0}-\overline{y})\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}}_{0}+2\hat{\beta}_{1}\underbrace{\sum_{i=1}^{n}\hat{% \varepsilon}_{i}x_{i1}}_{0}+\cdots+2\hat{\beta}_{p}\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}x_{ip}}_{0}\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}=\mathrm{ESS}+\mathrm{RSS}\\ \end{aligned}$

NTCIR12-MathWiki-20rate: 3

$r=r_{xy}=\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum^{n}_{i% =1}(x_{i}-\bar{x})^{2}}\sqrt{\sum^{n}_{i=1}(y_{i}-\bar{y})^{2}}}$

NTCIR12-MathWiki-20rate: 3

$r=\frac{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{s% _{X}s_{Y}}$

NTCIR12-MathWiki-20rate: 1

$s_{n-1}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}.$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle Var(\beta_{1})&\displaystyle=\frac{\sigma^{2}}{% \sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}.\end{aligned}$

NTCIR12-MathWiki-20rate: 0

$r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^{n}y}{\sum_{i=1}^{n}x}$

NTCIR12-MathWiki-20rate: 1

$\widehat{\beta_{1}}=\frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{% x})^{2}}\,\text{ and }\hat{\beta_{0}}=\bar{y}-\widehat{\beta_{1}}\bar{x}$

NTCIR12-MathWiki-20rate: 2

$\displaystyle\hat{\beta}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{% \sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}=\frac{\overline{xy}-\bar{x}\bar{y}}{% \overline{x^{2}}-\bar{x}^{2}}=\frac{\operatorname{Cov}[x,y]}{\operatorname{Var% }[x]}=r_{xy}\frac{s_{y}}{s_{x}},$

NTCIR12-MathWiki-20rate: 0

$\mathbf{Q}={1\over{N-1}}\sum_{i=1}^{N}(\mathbf{x}_{i}-\mathbf{\bar{x}})(% \mathbf{x}_{i}-\mathbf{\bar{x}})^{\mathrm{T}},$

NTCIR12-MathWiki-20rate: 1

$q_{jk}=\frac{\sum_{i=1}^{N}w_{i}}{\left(\sum_{i=1}^{N}w_{i}\right)^{2}-\sum_{i% =1}^{N}w_{i}^{2}}\sum_{i=1}^{N}w_{i}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik% }-\bar{x}_{k}\right).$

NTCIR12-MathWiki-20rate: 0

$S_{a}=\frac{E[R_{a}-R_{b}]}{\sigma_{a}}=\frac{E[R_{a}-R_{b}]}{\sqrt{\mathrm{% var}[R_{a}-R_{b}]}},$

NTCIR12-MathWiki-20rate: 1

$\displaystyle\frac{\partial\,\mathrm{SSE}\left(\hat{\alpha},\hat{\beta}\right)% }{\partial\hat{\beta}}=-2\sum_{i=1}^{n}\left[\left(y_{i}-\bar{y}\right)-\hat{% \beta}\left(x_{i}-\bar{x}\right)\right]\left(x_{i}-\bar{x}\right)=0$

NTCIR12-MathWiki-20rate: 1

$\displaystyle\hat{\beta}=\frac{}{}\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}% \right)\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2% }}=\frac{Cov\left(x,y\right)}{Var\left(x\right)}$

NTCIR12-MathWiki-20rate: 2

$b_{1}=\frac{m_{3}}{s^{3}}=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})% ^{3}}{\left[\tfrac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right]^{3/2}}\ ,$

NTCIR12-MathWiki-20rate: 1

$\lambda=\frac{\operatorname{E}(V)}{\operatorname{stdev}(V)}=\frac{\sum_{i=1}^{% t}c_{i}\mu_{i}}{\sqrt{\,\text{Var}(\sum_{i=1}^{t}c_{i}G_{i})}}=\frac{\sum_{i=1% }^{t}c_{i}\mu_{i}}{\sqrt{\sum_{i=1}^{t}c_{i}^{2}\sigma_{i}^{2}+2\sum_{i=1}^{t}% \sum_{j=i}c_{i}c_{j}\sigma_{ij}}}$

NTCIR12-MathWiki-20rate: 1

$\hat{\sigma}=\sqrt{\frac{1}{N-1.5}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}},$

NTCIR12-MathWiki-20rate: 1

$SE_{\widehat{\beta}}=\frac{\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}(y_{i}-\widehat{y}% _{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}}$

NTCIR12-MathWiki-20rate: 1

$\operatorname{var}(\widehat{\varepsilon}_{i})=\sigma^{2}\left(1-\frac{1}{n}-% \frac{(x_{i}-\bar{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\bar{x})^{2}}\right).$

NTCIR12-MathWiki-20rate: 1

$\begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}\tfrac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-(x_{i}-\theta)^{2}/(2\sigma^{2})}=(% 2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}(x_{i}-\theta)^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}((x_{i}-\overline{x}% )-(\theta-\overline{x}))^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,\exp\left({-1\over 2\sigma^{2}}\left(% \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}+\sum_{i=1}^{n}(\theta-\overline{x})^{2}% -2\sum_{i=1}^{n}(x_{i}-\overline{x})(\theta-\overline{x})\right)\right).\end{aligned}$

NTCIR12-MathWiki-20rate: 1

$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\,,$

NTCIR12-MathWiki-20rate: 3

$\mathrm{sim}(d_{j},q)=\frac{\mathbf{d_{j}}\cdot\mathbf{q}}{\left\|\mathbf{d_{j% }}\right\|\left\|\mathbf{q}\right\|}=\frac{\sum_{i=1}^{N}w_{i,j}w_{i,q}}{\sqrt% {\sum_{i=1}^{N}w_{i,j}^{2}}\sqrt{\sum_{i=1}^{N}w_{i,q}^{2}}}$

NTCIR12-MathWiki-20rate: 3

$NCC(\mu,\mathcal{I_{F}},\mathcal{I_{M}})=\dfrac{\sum_{x_{i}\in\Omega_{F}}\left% (\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)\left(\mathcal{I_{M}}% ({T}_{\mu}(x_{i}))-\overline{\mathcal{I_{M}}}\right)}{\sqrt{\sum_{x_{i}\in% \Omega_{F}}\left(\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)^{2}% \sum_{x_{i}\in\Omega_{F}}\left(\mathcal{I_{M}}({T}_{\mu}(x_{i}))-\overline{% \mathcal{I_{M}}}\right)^{2}}}$

NTCIR12-MathWiki-3rate: 2

$m_{Moon}=0.25+2.5\log_{10}{\left(\frac{3}{2}0.00257^{2}\right)}=-12.26\!\,$

NTCIR12-MathWiki-3rate: 3

$m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{f_{\nu}}{\,\text{Jy}}\right)+8.90.$

NTCIR12-MathWiki-3rate: 0

$\ C_{M}=-0.5\pi(A_{0}+A_{1}-A_{2}/2)$

NTCIR12-MathWiki-3rate: 0

$\textrm{cr}(K_{n})=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor% \frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor% \frac{n-3}{2}\right\rfloor.$

NTCIR12-MathWiki-3rate: 0

$\frac{m}{n}=-2eV_{1}\left(\frac{t}{f}\right)^{2}$

NTCIR12-MathWiki-3rate: 3

$\,\text{Attenuation (dB)}=10\times\log_{10}\left(\frac{\,\text{Input intensity% (W)}}{\,\text{Output intensity (W)}}\right)$

NTCIR12-MathWiki-3rate: 0

$\lfloor\log_{2}(0)\rfloor=-1$

NTCIR12-MathWiki-3rate: 1

$\lfloor\log_{2}N\rfloor$

NTCIR12-MathWiki-3rate: 0

$X=\frac{1}{2}\left[\left(\frac{T}{\beta}\right)^{0.5}-\left(\frac{T}{\beta}% \right)^{-0.5}\right]$

NTCIR12-MathWiki-3rate: 1

$x=\lfloor\log_{2}x\rfloor+1$

NTCIR12-MathWiki-3rate: 1

$r=f_{2}(\theta)-f_{1}(\theta)$

NTCIR12-MathWiki-3rate: 0

$\mathbf{P}=g(\mathbf{a})-g(\mathbf{b})$

NTCIR12-MathWiki-3rate: 0

$\alpha_{s}=\arctan\left(\frac{b}{R}\right)-\arctan\left(\frac{b}{R+r_{s}}\right)$

NTCIR12-MathWiki-3rate: 1

$k=\lfloor\log_{2}n\rfloor$

NTCIR12-MathWiki-3rate: 1

$-\log(x)$

NTCIR12-MathWiki-3rate: 1

$-\log(t)\,$

NTCIR12-MathWiki-3rate: 1

$\lfloor\log_{2}(x)\rfloor+2\lfloor\log_{2}(\lfloor\log_{2}(x)\rfloor+1)\rfloor+1$

NTCIR12-MathWiki-3rate: 1

$2\lfloor\log_{2}(x)\rfloor+1$

NTCIR12-MathWiki-3rate: 0

$d=-\left\lfloor\frac{m}{2}\right\rfloor$

NTCIR12-MathWiki-3rate: 1

$\gamma=1-\sum_{k=2}^{\infty}(-1)^{k}\frac{\lfloor\log_{2}k\rfloor}{k+1}.$

NTCIR12-MathWiki-3rate: 1

$\lfloor\log_{2}n\rfloor$

NTCIR12-MathWiki-3rate: 0

$206.835-1.015\left(\frac{\,\text{total words}}{\,\text{total sentences}}\right% )-84.6\left(\frac{\,\text{total syllables}}{\,\text{total words}}\right).$

NTCIR12-MathWiki-3rate: 0

$\left\lfloor\frac{m}{n}\right\rfloor+\left\lfloor\frac{2m}{n}\right\rfloor+% \dots+\left\lfloor\frac{(n-1)m}{n}\right\rfloor=\left\lfloor\frac{n}{m}\right% \rfloor+\left\lfloor\frac{2n}{m}\right\rfloor+\dots+\left\lfloor\frac{(m-1)n}{% m}\right\rfloor.$

NTCIR12-MathWiki-3rate: 1

$W=\phi(b)-\phi(a)$

NTCIR12-MathWiki-3rate: 1

$\log\left(x\right)=-\log\left(\frac{1}{x}\right)$

NTCIR12-MathWiki-3rate: 1

$\log\left(x\right)=-\log\left(\frac{1}{x}\right)$

NTCIR12-MathWiki-3rate: 1

$\Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s)$

NTCIR12-MathWiki-3rate: 0

$\lfloor n/2-1\rfloor=m$

NTCIR12-MathWiki-3rate: 1

$M=\left\lfloor\frac{-1}{\log_{2}(1-p)}\right\rfloor$

NTCIR12-MathWiki-3rate: 0

$\mu_{a}={{\mu_{0}}{{(1-0.5kH)}^{-2}}}$

NTCIR12-MathWiki-3rate: 2

$\mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)$

NTCIR12-MathWiki-3rate: 2

$\mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)$

NTCIR12-MathWiki-3rate: 2

$\log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)\,$

NTCIR12-MathWiki-3rate: 1

$f(n)=2(n-2^{\lfloor\log_{2}(n)\rfloor})+1$

NTCIR12-MathWiki-3rate: 0

$\mathbf{\delta}=\hat{\mathbf{h}}(n)-\mathbf{h}(n)$

NTCIR12-MathWiki-3rate: 0

$y=px-f^{\star}(p)$

NTCIR12-MathWiki-3rate: 0

$r=\lfloor 0.5+\mu_{2,l}\rfloor=4$

NTCIR12-MathWiki-3rate: 1

$\ell=\lfloor\log_{2}(\operatorname{depth}(v))\rfloor$

NTCIR12-MathWiki-3rate: 0

$R_{T}=f(x)-p(x)\,\!$

NTCIR12-MathWiki-3rate: 1

$2-\log_{2}(\sqrt{2})=\frac{3}{2}$

NTCIR12-MathWiki-3rate: 1

$\frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}$

NTCIR12-MathWiki-3rate: 2

$\log_{b}\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right)=\log_{b}(x)-\log_% {b}(y)$

NTCIR12-MathWiki-3rate: 2

$\log_{2}\!\left(\frac{1}{2}\right)=-1,\,$

NTCIR12-MathWiki-3rate: 1

$\log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)$

NTCIR12-MathWiki-3rate: 2

$m_{\rm star}=m_{\odot}-2.5\log_{10}\left[\frac{L_{\rm star}}{L_{\odot}}\left(% \frac{d_{\odot}}{d_{\rm star}}\right)^{2}\right]$

NTCIR12-MathWiki-3rate: 2

$A=-\log(\frac{V_{dc}}{V_{ex}})$

NTCIR12-MathWiki-3rate: 2

$M=\log_{2}(R/G)=\log_{2}(R)-\log_{2}(G)$

NTCIR12-MathWiki-3rate: 1

$[\mathrm{Fe}/\mathrm{H}]=\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}% }\right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}% }}\right)_{\mathrm{sun}}}$

NTCIR12-MathWiki-3rate: 3

$d=69+12\log_{2}\left(\frac{f}{440\ \mathrm{Hz}}\right).\,$

NTCIR12-MathWiki-3rate: 1

$1+\lfloor\log_{2}N\rfloor$

NTCIR12-MathWiki-3rate: 2

$L=\left\lceil\frac{\lfloor\log_{2}n\rfloor+1}{2}\right\rceil$

NTCIR12-MathWiki-3rate: 0

$0=-\frac{\mbox{d}~{}P}{\mbox{d}~{}x}+\mu\left(\frac{\mbox{d}~{}^{2}u}{\mbox{d}% ~{}y^{2}}\right)$

NTCIR12-MathWiki-3rate: 0

$\alpha=\arctan\left({R\over L}\right)-\arccos\left({\sqrt{L^{2}+R^{2}}\over 2% \rho}\right)$

NTCIR12-MathWiki-3rate: 0

$\left\lfloor\frac{(n-1)}{m}\right\rfloor+1=\left\lfloor\frac{(7-1)}{4}\right% \rfloor+1=\left\lfloor\frac{6}{4}\right\rfloor+1=1+1=2$

NTCIR12-MathWiki-3rate: 1

$f(n)=n+\left\lfloor\log_{2}\left(n+\log_{2}n\right)\right\rfloor.$

NTCIR12-MathWiki-3rate: 0

$W_{C}=U(B)-U(A),$

NTCIR12-MathWiki-3rate: 1

$-\log_{10}\left(\frac{1+2d}{365}\right)$

NTCIR12-MathWiki-3rate: 1

$\mathrm{length}(c_{k})\approx-\log_{2}\left(p_{k}\right)$

NTCIR12-MathWiki-3rate: 1

$\Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s),$

NTCIR12-MathWiki-3rate: 0

$q=\left\lceil y-0.5\right\rceil=-\left\lfloor-y+0.5\right\rfloor\,$

NTCIR12-MathWiki-3rate: 1

$\lfloor\log_{2}(n)\rfloor$

NTCIR12-MathWiki-3rate: 0

$H_{HPF}(f)=1-\mathrm{rect}\left(\frac{f}{2B_{H}}\right).$

NTCIR12-MathWiki-3rate: 2

$\log(x/y)=\log(x)-\log(y)$

NTCIR12-MathWiki-3rate: 2

$L_{N}=40+10\log_{2}(N)$

NTCIR12-MathWiki-3rate: 2

$L_{W}=L_{p}+10\log_{10}\!\left(\frac{4\pi r^{2}}{A_{0}}\right)\!~{}\mathrm{dB},$

NTCIR12-MathWiki-3rate: 2

$L_{p_{2}}=L_{p_{1}}+20\log_{10}\!\left(\frac{r_{1}}{r_{2}}\right)\!~{}\mathrm{% dB}.$

NTCIR12-MathWiki-3rate: 0

$e=H(x)-H(\hat{x})$

NTCIR12-MathWiki-3rate: 0

$\dot{e}_{2}=h_{3}(\hat{x})-m_{2}(\hat{x})\operatorname{sgn}(e_{2})$

NTCIR12-MathWiki-3rate: 1

$=\lim_{N\to\infty}Nb\log_{2}\left(\frac{A}{NW}+1\right)$

NTCIR12-MathWiki-3rate: 1

$\gamma_{1}\,=\,\frac{\ln 2}{2}\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}\,\lfloor% \log_{2}{k}\rfloor\cdot\big(2\log_{2}{k}-\lfloor\log_{2}{2k}\rfloor\big)$

NTCIR12-MathWiki-3rate: 1

$r_{\infty}=ms-\deg(f)$

NTCIR12-MathWiki-3rate: 2

$1+\lfloor\log_{2}n\rfloor$

NTCIR12-MathWiki-3rate: 0

$c_{2}I=c_{1}E-S_{e}^{-1}\left(\frac{E}{k_{e}-r_{e}E}\right)+P$

NTCIR12-MathWiki-3rate: 4

$N=\left\lfloor 0.5-\log_{2}\left(\frac{\,\text{Frequency of this item}}{\,% \text{Frequency of most common item}}\right)\right\rfloor$

NTCIR12-MathWiki-3rate: 0

$N=\lfloor u^{2}\rfloor$

NTCIR12-MathWiki-4rate: 1

$\frac{\partial}{\partial t}(\rho_{0}+\rho_{0}s)+\frac{\partial}{\partial x}(% \rho_{0}u+\rho_{0}su)=0$

NTCIR12-MathWiki-4rate: 2

$\mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}$

NTCIR12-MathWiki-4rate: 0

$\mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}=(\Omega_{\mathbf{% k}}^{\star}\,p_{\mathbf{k}}-\Omega_{\mathbf{k}}\,p_{\mathbf{k}}^{\star})+% \mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}|_{\,\text{corr}}\,,$

NTCIR12-MathWiki-4rate: 2

$symbol{\nabla\times}\left(symbol{\nabla\times B}\right)=\mu_{0}\epsilon_{0}% \frac{\partial}{\partial t}symbol{\nabla\times E}\ .$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 1

$\nabla^{2}\mathbf{E}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}$

NTCIR12-MathWiki-4rate: 1

$\nabla\times\mathbf{E}=\hat{\mathbf{k}}\times\mathbf{E}_{0}f^{\prime}\left(% \hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=-\frac{\partial\mathbf{B}}{% \partial t}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t% }\qquad\quad\ (4)$

NTCIR12-MathWiki-4rate: 3

$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}},\quad\nabla\times\mathbf{B}-% \frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=\mu_{0}\mathbf{J}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{j}$

NTCIR12-MathWiki-4rate: 1

$\nabla\cdot\mathbf{B}=\mu_{0}\rho_{m}$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\frac{1}{\epsilon_{0}c^{2}}\mathbf{J}+\frac{1}{c^{2}}% \frac{\partial\mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 0

$+\underbrace{E_{\mathrm{sig}}E_{\mathrm{LO}}\cos((\omega_{\mathrm{sig}}-\omega% _{\mathrm{LO}})t+\varphi)}_{beat\;component}.$

NTCIR12-MathWiki-4rate: 2

$\vec{\nabla}\times\vec{B}=\mu_{0}\vec{J},$

NTCIR12-MathWiki-4rate: 1

$\nabla^{2}\varphi+{{\partial}\over\partial t}\left(\nabla\cdot\mathbf{A}\right% )=-{\rho\over\varepsilon_{0}}\,,$

NTCIR12-MathWiki-4rate: 4

$\nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}$

NTCIR12-MathWiki-4rate: 2

$\nabla\cdot\vec{E}={\rho\over\epsilon_{0}}\qquad\nabla\times\vec{B}-\epsilon_{% 0}\mu_{0}{\partial\vec{E}\over\partial t}=\mu_{0}\vec{J}\qquad\nabla\times\vec% {E}+{\partial\vec{B}\over\partial t}=0\qquad\nabla\cdot\vec{B}=0$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t},$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}$

NTCIR12-MathWiki-4rate: 4

$\nabla\times\mathbf{B}=\mu\mathbf{J}+\mu\epsilon\frac{\partial\mathbf{E}}{% \partial t}.$

NTCIR12-MathWiki-4rate: 1

$\mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}$

NTCIR12-MathWiki-4rate: 2

$\mu_{0}\mathbf{J}=\nabla\times\mathbf{B}$

NTCIR12-MathWiki-4rate: 2

$\mu_{0}\mathbf{J}=\nabla\times\mathbf{B}.$

NTCIR12-MathWiki-4rate: 3

$\mathbf{\nabla}\cdot\mathbf{A}^{\prime}=-\mu_{0}\varepsilon_{0}\frac{\partial% \varphi^{\prime}}{\partial t}$

NTCIR12-MathWiki-4rate: 4

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)$

NTCIR12-MathWiki-4rate: 4

$\nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=% \mu_{0}\mathbf{J}$

NTCIR12-MathWiki-4rate: 3

$\mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}$

NTCIR12-MathWiki-4rate: 4

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\epsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 4

$\mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}+\underbrace{\mu_{0}\epsilon_% {0}\frac{\partial}{\partial t}\mathbf{E}}_{\mathrm{Maxwell^{\prime}s\ term}}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\vec{B}=\mu_{0}\vec{J}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\vec{B}=\mu_{0}\vec{J}$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\vec{B}=\mu_{0}\vec{J}$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\frac{1}{c^{2}}\left(\frac{1}{\epsilon_{0}}\mathbf{J}+% \frac{\partial\mathbf{E}}{\partial t}\right)$

NTCIR12-MathWiki-4rate: 1

$\nabla\cdot\mathbf{S}+\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{% \partial t}+\frac{\mathbf{B}}{\mu_{0}}\cdot\frac{\partial\mathbf{B}}{\partial t% }+\mathbf{J}\cdot\mathbf{E}=0,$

NTCIR12-MathWiki-4rate: 0

$\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 0

$\nabla^{2}E-\mu_{0}\,\varepsilon_{0}\,\frac{\partial^{2}E}{\partial t^{2}}=0.$

NTCIR12-MathWiki-4rate: 0

$I=\underbrace{\tau\int_{-\infty}^{0}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x% }_{I_{1}}+\underbrace{\tau\int_{0}^{\infty}\frac{H(\mu+\tau x)}{e^{x}+1}\,% \mathrm{d}x}_{I_{2}}\,.$

NTCIR12-MathWiki-4rate: 3

$\nabla\times\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{% \partial t}=\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}$

NTCIR12-MathWiki-4rate: 0

${\underbrace{\partial h\over\partial t}}_{\begin{smallmatrix}\,\text{Change}\\ \,\text{in mass}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\overline{hu}\over% \partial x}+{\partial\overline{hv}\over\partial y}}_{\begin{smallmatrix}\,% \text{Total spatial}\\ \,\text{variation of}\\ \,\text{x,y mass fluxes}\end{smallmatrix}}=0$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B}=\mu_{0}\mathbf{j}$

NTCIR12-MathWiki-4rate: 2

$\nabla^{2}\mathbf{E}-\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}=0.$

NTCIR12-MathWiki-4rate: 1

$\nabla\times\hat{\mathbf{B}}=\mu_{0}\hat{\mathbf{J}}+\mathrm{i}\mu_{0}% \varepsilon_{0}\omega\hat{\mathbf{E}}\quad\Rightarrow\quad-\frac{B^{r}}{r}+% \frac{\mathrm{d}B^{(1)}}{\mathrm{d}r}+\frac{B^{(1)}}{r}=\mu_{0}J+\mathrm{i}% \omega\mu_{0}\varepsilon_{0}E$

NTCIR12-MathWiki-4rate: 2

$\nabla\times\mathbf{B_{1}}=\mu_{0}\mathbf{J_{1}}-i\omega\epsilon_{0}\mu_{0}% \mathbf{E_{1}}$

NTCIR12-MathWiki-5rate: 1

$1+\frac{1}{4+\cfrac{1}{1+\cfrac{1}{18+\cfrac{1}{\ddots\qquad{}}}}}$

NTCIR12-MathWiki-5rate: 1

$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}\!$

NTCIR12-MathWiki-5rate: 4

$1+\cfrac{1}{2+\cfrac{1}{5+\cfrac{1}{5+\cfrac{1}{4+\ddots}}}}$

NTCIR12-MathWiki-5rate: 1

$\cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{9+\ddots}}}}}$

NTCIR12-MathWiki-5rate: 2

$\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}% {1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}$

NTCIR12-MathWiki-5rate: 1

$\displaystyle\pi=2+\cfrac{4}{3+\cfrac{1\cdot 3}{4+\cfrac{3\cdot 5}{4+\cfrac{5% \cdot 7}{4+\ddots}}}}$

NTCIR12-MathWiki-5rate: 1

$\pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}.\,$

NTCIR12-MathWiki-5rate: 1

$e^{2}=1+\cfrac{4}{0+\cfrac{2^{2}}{6+\cfrac{2^{2}}{10+\cfrac{2^{2}}{14+\ddots\,% }}}}=7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\ddots\,}}}}$

NTCIR12-MathWiki-5rate: 2

$e=2+\cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf% {4}+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}=1+\cfrac{1}{\mathbf{0}+\cfrac{1}{1+% \cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf{4}+% \cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}}}.$

NTCIR12-MathWiki-5rate: 1

$e=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{2% 2+\cfrac{1}{26+\ddots\,}}}}}}}.$

NTCIR12-MathWiki-5rate: 1

$\varphi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\;\;\ddots\,}}}$

NTCIR12-MathWiki-5rate: 2

$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+{}\ddots}}}}}$

NTCIR12-MathWiki-5rate: 3

$x=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots\,}}}}$

NTCIR12-MathWiki-5rate: 1

$\log 2=\log(1+1)=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+% \cfrac{3}{2+\ddots}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2}}{15-\cfrac{3% ^{2}}{21-\ddots}}}}$

NTCIR12-MathWiki-5rate: 2

$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}$

NTCIR12-MathWiki-5rate: 4

$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{5+\cfrac{1}{1+\cfrac{1}{4+\ddots}}}}}$

NTCIR12-MathWiki-5rate: 1

$\pi={3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\cfrac{7^{2}}{6+\ddots% \,}}}}}$

NTCIR12-MathWiki-5rate: 1

$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\cfrac{4^{2}}{% 9+\ddots}}}}}$

NTCIR12-MathWiki-5rate: 1

$\pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}\,$

NTCIR12-MathWiki-5rate: 2

$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}}=2+% \cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}$

NTCIR12-MathWiki-5rate: 1

$\tfrac{1}{1+\tfrac{1}{0+\tfrac{1}{8+\tfrac{1}{4+\tfrac{1}{1+\tfrac{1}{0+1{/% \cdots}}}}}}}$

NTCIR12-MathWiki-5rate: 2

$\ln 2=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+% \cfrac{3}{7+\cfrac{4}{2+\ddots}}}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2% }}{15-\cfrac{3^{2}}{21-\ddots}}}}$

NTCIR12-MathWiki-5rate: 1

$\sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots% \,}}}}}.$

NTCIR12-MathWiki-5rate: 1

$\frac{577}{408}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+% \cfrac{1}{2+\cfrac{1}{2}}}}}}}.$

NTCIR12-MathWiki-5rate: 1

$\pi=3+\textstyle\frac{1}{7+\textstyle\frac{1}{15+\textstyle\frac{1}{1+% \textstyle\frac{1}{292+\textstyle\frac{1}{1+\textstyle\frac{1}{1+\textstyle% \frac{1}{1+\ddots}}}}}}}$

NTCIR12-MathWiki-5rate: 1

$a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}},$

NTCIR12-MathWiki-5rate: 1

$2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}$

NTCIR12-MathWiki-5rate: 1

$\tanh\frac{1}{2}=\frac{e-1}{e+1}=0+\cfrac{1}{2+\cfrac{1}{6+\cfrac{1}{10+\cfrac% {1}{14+\cfrac{1}{\ddots}}}}}$

NTCIR12-MathWiki-5rate: 2

$x=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}}=% \sqrt{2}.\,$

NTCIR12-MathWiki-5rate: 1

$\sqrt{11}=3+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\ddots% }}}}}$

NTCIR12-MathWiki-5rate: 1

$1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}$

NTCIR12-MathWiki-5rate: 2

$1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\ddots}}}}}$

NTCIR12-MathWiki-5rate: 1

$2+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\ddots}}}}$

NTCIR12-MathWiki-5rate: 3

${1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}}$

NTCIR12-MathWiki-5rate: 1

$\frac{4}{\pi}=1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\cfrac{9^{2}}{2+\ddots}}}}}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{92}U+{}^{1}_{0}n\ \xrightarrow{\ }\ {}^{239}_{92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ {}^{239}_{93}Np\ \xrightarrow[2.3\ days]{% \beta^{-}}\ {}^{239}_{94}Pu\ \xrightarrow[2.4\cdot 10^{4}\ years]{\alpha}}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{226}_{\ 88}Ra\ +\ ^{1}_{0}n\ \longrightarrow\ ^{227}_{\ 88}Ra\ % \xrightarrow[42.2\ min]{\beta^{-}}\ ^{227}_{\ 89}Ac}$

NTCIR12-MathWiki-6rate: 4

$\mathrm{~{}^{238}_{92}U}\rightarrow\mathrm{~{}^{234}_{90}Th}+{\alpha}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}_{20}^{40}Ca}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{22}^{44}Ti% }+\gamma$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 0

$B+X\rightarrow Y+D$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{10}_{\ 5}B\ \longrightarrow\ ^{242}_{\ 97}Bk\ +% \ 6\ ^{1}_{0}n\quad;\quad^{232}_{\ 90}Th\ +\ ^{15}_{\ 7}N\ \longrightarrow\ ^{% 242}_{\ 97}Bk\ +\ 5\ ^{1}_{0}n}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 1

$\mathrm{Glucose}+\mathrm{Alkaline\ copper\ tartarate}\xrightarrow{\mathrm{% Reduction}}\mathrm{Cuprous\ oxide}$

NTCIR12-MathWiki-6rate: 0

$B+X\rightarrow Y+D$

NTCIR12-MathWiki-6rate: 1

$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow[-2\ \beta^{-}]{+\ 6\ (n,\gamma)}\ ^{244% }_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 3

$\,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}\to\ \ no\ atoms$

NTCIR12-MathWiki-6rate: 2

$\,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}$

NTCIR12-MathWiki-6rate: 0

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

NTCIR12-MathWiki-6rate: 0

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{+\ 15n,7\beta^{-}}\ ^{253}_{\ 99}Es}$

NTCIR12-MathWiki-6rate: 0

$n+p\rightarrow d+\gamma$

NTCIR12-MathWiki-6rate: 0

$\gamma_{\rm CMB}+p\rightarrow\Delta^{+}\rightarrow n+\pi^{+}.$

NTCIR12-MathWiki-6rate: 0

$\nu+\bar{\nu}\rightarrow Z\rightarrow\,\text{hadrons}$

NTCIR12-MathWiki-6rate: 0

$i:A\to B\to D$

NTCIR12-MathWiki-6rate: 0

$e^{-}+p\to\nu_{e}+n$

NTCIR12-MathWiki-6rate: 1

$\mathrm{{}^{7}_{4}Be}+\mathrm{e{}^{-}}\ \xrightarrow{\ \mathrm{53.22d}}\ % \mathrm{{}^{7}_{3}Li}$

NTCIR12-MathWiki-6rate: 4

$\,{}^{238}_{92}\mathrm{U}\ +\,^{24}_{12}\mathrm{Mg}\to\,^{259}_{104}\mathrm{Rf% }\ +3\,^{1}_{0}\mathrm{n}$

NTCIR12-MathWiki-6rate: 1

$\displaystyle{M\rightarrow\mathfrak{k}^{*}\rightarrow\mathfrak{t}^{*}.}$

NTCIR12-MathWiki-6rate: 2

${}^{2}_{1}\mathrm{D}+\,^{3}_{1}\mathrm{T}\rightarrow\,^{4}_{2}\mathrm{He}\left% (3.5\,\mathrm{MeV}\right)+\,^{1}_{0}\mathrm{n}\left(14.1\,\mathrm{MeV}\right)$

NTCIR12-MathWiki-6rate: 0

$\pi^{+}\to W^{*}\to\mu^{+}\nu_{\mu}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{235}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{236}_{\ 92}U_{m}\ % \xrightarrow[120\ ns]{}\ ^{236}_{\ 92}U\ +\ \gamma}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{236}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{237}_{\ 92}U\ % \xrightarrow[6.75\ d]{\beta^{-}}\ ^{237}_{\ 93}Np}$

NTCIR12-MathWiki-6rate: 1

${{\nu}_{e}}+n\to p+e^{-}$

NTCIR12-MathWiki-6rate: 0

$\vec{n}+p\to d+\gamma$

NTCIR12-MathWiki-6rate: 0

$A+n\to B^{*}+\gamma$

NTCIR12-MathWiki-6rate: 2

$\rm Succinate+Q\rightarrow Fumarate+QH_{2}\!$

NTCIR12-MathWiki-6rate: 0

$\nu_{e}+n\rightarrow e^{-}+p$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U\ +\ ^{2}_{1}D\ \longrightarrow\ ^{238}_{\ 93}Np\ +\ 2% \ ^{1}_{0}n\quad;\quad^{238}_{\ 93}Np\ \xrightarrow[2.117\ d]{\beta^{-}}\ ^{23% 8}_{\ 94}Pu}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{{}^{232}_{\ 90}Th\ +\ ^{1}_{0}n\ \longrightarrow\ ^{233}_{\ 90}Th\ % \xrightarrow[22.3\ min]{\beta^{-}}\ ^{233}_{\ 91}Pa\ \xrightarrow[26.967\ d]{% \beta^{-}}\ ^{233}_{\ 92}U}$

NTCIR12-MathWiki-6rate: 0

$U\overset{j}{\to}V\overset{g}{\to}Y$

NTCIR12-MathWiki-6rate: 0

$A+\gamma\rightarrow A^{*}$

NTCIR12-MathWiki-6rate: 0

$T_{0}+T_{0}\rightarrow S^{*}+S_{0}+\,\text{phonons}$

NTCIR12-MathWiki-6rate: 2

${}_{\ 90}^{232}\mathrm{Th}+\mathrm{n}\rightarrow{}_{\ 90}^{233}\mathrm{Th}+% \gamma\ \xrightarrow{\beta^{-}}\ {}_{\ 91}^{233}\mathrm{Pa}\ \xrightarrow{% \beta^{-}}\ {}_{\ 92}^{233}\mathrm{U}$

NTCIR12-MathWiki-6rate: 1

$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}$

NTCIR12-MathWiki-6rate: 2

$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{232}\mathrm{U}+2\mathrm{n}$

NTCIR12-MathWiki-6rate: 1

$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{231}\mathrm{Th}+2% \mathrm{n}\xrightarrow{\beta^{-}}{}_{\ 91}^{231}\mathrm{Pa}+\mathrm{n}% \rightarrow{}_{\ 91}^{232}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{232}% \mathrm{U}$

NTCIR12-MathWiki-6rate: 0

$0\to U\to M\to V\to 0$

NTCIR12-MathWiki-6rate: 3

$\mathrm{{}^{238}_{\ 92}U+\,^{1}_{0}n\;\rightarrow\;^{239}_{\ 92}U\;\rightarrow% \;^{239}_{\ 93}Np+\beta\;\rightarrow\;^{239}_{\ 94}Pu+\beta}$

NTCIR12-MathWiki-6rate: 4

$\,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}$

NTCIR12-MathWiki-6rate: 3

$\,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}$

NTCIR12-MathWiki-6rate: 4

${}_{\ 92}^{238}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{239}\mathrm{U}+% \gamma\xrightarrow{\beta^{-}}{}_{\ 93}^{239}\mathrm{Np}\xrightarrow{\beta^{-}}% {}_{\ 94}^{239}\mathrm{Pu}$

NTCIR12-MathWiki-6rate: 1

$U+W$

NTCIR12-MathWiki-6rate: 0

$U+W$

NTCIR12-MathWiki-6rate: 0

$U+W$

NTCIR12-MathWiki-6rate: 0

$U+W$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to V^{\prime}\to V\to V^{\prime\prime}\to 0,$

NTCIR12-MathWiki-7rate: 2

$0\to VP\to TP\xrightarrow{d\pi}\pi^{*}TM\to 0$

NTCIR12-MathWiki-7rate: 2

$\ 0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow\ker h\rightarrow H^{n}(C;G)\stackrel{h}{\rightarrow}\,\text{Hom}(% H_{n}(C),G)\rightarrow 0.$

NTCIR12-MathWiki-7rate: 2

$0\to\mathbb{Z}\to\mathcal{O}\to\mathcal{O}^{*}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to 2\pi i\mathbb{Z}\to\mathbf{O}\xrightarrow{\exp}\mathbf{O}^{*}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow M^{\prime}\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0$

NTCIR12-MathWiki-7rate: 1

$\cdots\to X^{-1}\xrightarrow{d^{-1}}X^{0}\xrightarrow{d^{0}}X^{1}\xrightarrow{% d^{1}}X^{2}\to\cdots$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 3

$0\to H\stackrel{i}{{}\to{}}G\stackrel{\pi}{{}\to{}}G/H\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to G^{p+1}\to G^{p}\to K^{p,*}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\xrightarrow{f}M\to M^{\prime\prime}\to 0,$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 4

$0\to G^{\wedge}\stackrel{\pi^{\wedge}}{\to}X^{\wedge}\stackrel{\imath^{\wedge}% }{\to}H^{\wedge}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow B\rightarrow Y\rightarrow A\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow B\rightarrow E^{\prime}\rightarrow A\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\to I\to L\to L^{\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to L\to M\to N\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow\mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$1\rightarrow K\stackrel{i}{\rightarrow}G\stackrel{\pi}{\rightarrow}H\rightarrow 1$

NTCIR12-MathWiki-7rate: 2

$1\to K\stackrel{i^{\prime}}{\rightarrow}G^{\prime}\stackrel{\pi^{\prime}}{% \rightarrow}H\rightarrow 1$

NTCIR12-MathWiki-7rate: 2

$0\;\rightarrow\;A\;\rightarrow\;B\;\rightarrow\;C\;\rightarrow\;0$

NTCIR12-MathWiki-7rate: 1

$0\to C^{0}\stackrel{d_{0}}{\longrightarrow}C^{1}\stackrel{d_{1}}{% \longrightarrow}C^{2}\stackrel{d_{2}}{\longrightarrow}\cdots\stackrel{d_{n-1}}% {\longrightarrow}C^{n}\longrightarrow 0.$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow\mathbb{R}\rightarrow X\rightarrow V\rightarrow 0.\,\!$

NTCIR12-MathWiki-7rate: 2

$0\to R\xrightarrow{\ d_{2}\ }R^{2}\xrightarrow{\ d_{1}\ }R\to 0,$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow M\rightarrow\mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow A^{\prime}\rightarrow A\rightarrow A^{\prime\prime}\rightarrow 0$

NTCIR12-MathWiki-7rate: 0

$G^{\wedge}$

NTCIR12-MathWiki-7rate: 0

$\pi\in G^{\wedge}$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0\,$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0\;$

NTCIR12-MathWiki-7rate: 2

$0\to R\xrightarrow{r-s}X\to Y\to 0$

NTCIR12-MathWiki-7rate: 0

$n^{\wedge}A$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow K\rightarrow X\rightarrow P^{\prime}\rightarrow 0,$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow\mathrm{Pic}(X)\rightarrow\mathbb{Z}\stackrel{\delta}{\rightarrow}% \mathrm{Br}(K)\rightarrow\mathrm{Br}(K)(X)\rightarrow 0\ .$

NTCIR12-MathWiki-7rate: 2

$0\ \rightarrow\mathcal{A}\ \stackrel{\phi}{\rightarrow}\ \mathcal{B}\ % \stackrel{\psi}{\rightarrow}\ \mathcal{C}\ \rightarrow\ 0$

NTCIR12-MathWiki-7rate: 2

$0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.$

NTCIR12-MathWiki-7rate: 3

$0\rightarrow A\stackrel{q}{\longrightarrow}B\stackrel{r}{\longrightarrow}C% \rightarrow 0\,$

NTCIR12-MathWiki-7rate: 2

$0\to X\to E\to Y\to 0\,$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

NTCIR12-MathWiki-7rate: 2

$0\to U\to M\to V\to 0$

NTCIR12-MathWiki-7rate: 3

$W\stackrel{f}{\ \to\ }X\stackrel{g}{\ \to\ }Y\stackrel{h}{\ \to\ }Z$

NTCIR12-MathWiki-7rate: 2

$0\to A\to B\to C\to 0$

NTCIR12-MathWiki-8rate: 3

$|a|=\begin{cases}a,&\mbox{if }~{}a\geq 0\\ -a,&\mbox{if }~{}a\leq 0\end{cases}\;$

NTCIR12-MathWiki-8rate: 0

$\lambda(n)=\begin{cases}\;\;\phi(n)&\,\text{if }n=2,3,4,5,7,9,11,13,17,19,23,2% 5,27,\dots\\ \tfrac{1}{2}\phi(n)&\,\text{if }n=8,16,32,64,\dots\end{cases}$

NTCIR12-MathWiki-8rate: 2

$f(x)=\begin{cases}\frac{1}{x}&\mbox{if }~{}x>0,\\ 5&\mbox{if }~{}x\leq 0.\end{cases}$

NTCIR12-MathWiki-8rate: 0

$a(n)=\begin{cases}\frac{2^{n+1}-2}{3},&\,\text{when }n\,\text{ is even,}\\ \frac{2^{n+1}-1}{3},&\,\text{when }n\,\text{ is odd.}\end{cases}$

NTCIR12-MathWiki-8rate: 1

$\log^{*}(n)=\begin{cases}0,&\,\text{if }n\leq 1\\ 1+\log^{*}(\log n),&\,\text{if }n>1\end{cases}$

NTCIR12-MathWiki-8rate: 2

${\left({{n}\atop{k}}\right)}=\begin{cases}n^{\underline{k}}/k!&\,\text{if }\ k% \leq\frac{n}{2}\\ n^{\underline{n-k}}/(n-k)!&\,\text{if }\ k>\frac{n}{2}\end{cases}.$

NTCIR12-MathWiki-8rate: 4

$Y=\begin{cases}0,&\mbox{if }~{}Y^{*}>0\\ 1,&\mbox{if }~{}Y^{*}<0.\end{cases}$

NTCIR12-MathWiki-8rate: 2

$Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}$

NTCIR12-MathWiki-8rate: 2

$Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}$

NTCIR12-MathWiki-8rate: 0

$\lambda(n)=\begin{cases}\;\;\varphi(n)&\mbox{if }~{}n=2,3,4,5,6,7,9,10,11,13,1% 4,17,19,22,23,25,26,27,29\dots\\ \tfrac{1}{2}\varphi(n)&\,\text{if }n=8,16,32,64,128,256\dots\end{cases}$

NTCIR12-MathWiki-8rate: 1

$T_{j}(x)U_{k}(x)=\begin{cases}\tfrac{1}{2}\left(U_{j+k}(x)+U_{k-j}(x)\right),&% \,\text{if }k\geq j-1.\\ \tfrac{1}{2}\left(U_{j+k}(x)-U_{j-k-2}(x)\right),&\,\text{if }k\leq j-2.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$V_{\,\text{out}}=\begin{cases}V_{\,\text{S}+}&\,\text{if }V_{1}>V_{2},\\ V_{\,\text{S}-}&\,\text{if }V_{1}<V_{2},\\ 0&\,\text{if }V_{1}=V_{2},\end{cases}$

NTCIR12-MathWiki-8rate: 0

$P^{*}=\frac{1}{2}$

NTCIR12-MathWiki-8rate: 1

$\begin{cases}x^{*}\log(x^{*})-x^{*}&\,\text{if }x^{*}>0\\ 0&\,\text{if }x^{*}=0\end{cases}$

NTCIR12-MathWiki-8rate: 2

$f^{\star}\left(x^{*}\right)=\begin{cases}0,&\left|x^{*}\right|\leq 1\\ \infty,&\left|x^{*}\right|>1.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$v_{i,j}=\begin{cases}n^{-\frac{1}{2}}&j=1\\ \sqrt{\frac{2}{n}}\cos(\frac{\pi(j-1)(i-\frac{1}{2})}{n})&otherwise\end{cases}$

NTCIR12-MathWiki-8rate: 1

$\zeta=\begin{cases}1&\mbox{if }~{}n\neq 0\\ 1/2&\mbox{if }~{}n=0\end{cases},\quad\xi=\begin{cases}1&\mbox{if }~{}m\neq 0\\ 1/2&\mbox{if }~{}m=0\end{cases}$

NTCIR12-MathWiki-8rate: 1

$c_{n}\ \stackrel{\mathrm{def}}{=}\ \begin{cases}\frac{A_{n}}{2i}e^{i\phi_{n}}=% \frac{1}{2}(a_{n}-ib_{n})&\,\text{for }n>0\\ \frac{1}{2}a_{0}&\,\text{for }n=0\\ c_{|n|}^{*}&\,\text{for }n<0.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$\psi(t)=\begin{cases}1&0\leq t<\frac{1}{2},\\ -1&\frac{1}{2}\leq t<1,\\ 0&\mbox{otherwise.}\end{cases}$

NTCIR12-MathWiki-8rate: 1

$H[n]=\begin{cases}0,&n<0,\\ 1/2,&n=0,\\ 1,&n>0,\end{cases}$

NTCIR12-MathWiki-8rate: 0

$f(x)=\begin{cases}1&-1\leq x<0\\ \frac{1}{2}&x=0\\ 1-x^{2}&\,\text{otherwise}\end{cases}$

NTCIR12-MathWiki-8rate: 2

$x_{+}^{\alpha}=\begin{cases}x^{\alpha}&\,\text{if }x>0\\ 0&\,\text{otherwise}\end{cases}$

NTCIR12-MathWiki-8rate: 2

$\log^{*}n:=\begin{cases}0&\mbox{if }~{}n\leq 1;\\ 1+\log^{*}(\log n)&\mbox{if }~{}n>1\end{cases}$

NTCIR12-MathWiki-8rate: 1

$\delta_{ij}=\begin{cases}0&\,\text{if }i\neq j,\\ 1&\,\text{if }i=j.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$f(t)=\begin{cases}t^{1/3}&\,\text{if }t>(\frac{6}{29})^{3}\\ \frac{1}{3}\left(\frac{29}{6}\right)^{2}t+\frac{4}{29}&\,\text{otherwise}\end{cases}$

NTCIR12-MathWiki-8rate: 1

$f^{-1}(t)=\begin{cases}t^{3}&\,\text{if }t>\tfrac{6}{29}\\ 3\left(\tfrac{6}{29}\right)^{2}\left(t-\tfrac{4}{29}\right)&\,\text{otherwise}% \end{cases}$

NTCIR12-MathWiki-8rate: 2

$f^{*}(x^{*})=\begin{cases}2x^{*}-4,&x^{*}<4\\ \frac{{x^{*}}^{2}}{4},&4\leqslant x^{*}\leqslant 6,\\ 3x^{*}-9,&x^{*}>6\end{cases}$

NTCIR12-MathWiki-8rate: 2

$\mbox{absolute margin of victory}~{}=\begin{cases}0;&w\leq\frac{c}{2}\\ w-\max\{r,\frac{c}{2}\};&w>\frac{c}{2}\end{cases}$

NTCIR12-MathWiki-8rate: 1

$Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{1\ast}>Y_{i}^{0\ast},\\ 0&\,\text{otherwise.}\end{cases}$

NTCIR12-MathWiki-8rate: 2

$\delta^{a}_{b}=\begin{cases}1&\mbox{if }~{}a=b,\\ 0&\mbox{if }~{}a\neq b.\end{cases}$

NTCIR12-MathWiki-8rate: 3

$\Pi(t)\ \stackrel{\,\text{def}}{=}\ \begin{cases}1&\,\text{if }|t|<\frac{1}{2}% ,\\ 0&\,\text{if }|t|>\frac{1}{2}.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$M(n)=\left\{\begin{matrix}n-10,&\mbox{if }~{}n>100\mbox{ }\\ M(M(n+11)),&\mbox{if }~{}n\leq 100\mbox{ }\end{matrix}\right.$

NTCIR12-MathWiki-8rate: 2

$f(x)=\begin{cases}1,&\mathrm{if}~{}|x|<\tfrac{1}{2},\\ 0,&\mathrm{otherwise.}\end{cases}$

NTCIR12-MathWiki-8rate: 1

$P_{ij}(\nu)=\left\{\begin{array}[]{cc}{1\over 4}+{3\over 4}e^{-4\nu/3}&\mbox{ % if }~{}i=j\\ {1\over 4}-{1\over 4}e^{-4\nu/3}&\mbox{ if }~{}i\neq j\end{array}\right.$

NTCIR12-MathWiki-8rate: 2

$w=\begin{cases}w_{min},&\mbox{if }~{}w_{min}\geq\;w(L)\\ w(L),&\mbox{if }~{}w_{min}\leq\;w(L)\end{cases}\,\!$

NTCIR12-MathWiki-8rate: 0

$\mathbf{1}_{A}(x)=\begin{cases}1&\,\text{if }x\in A,\\ 0&\,\text{if }x\notin A.\end{cases}$

NTCIR12-MathWiki-8rate: 2

$Y_{1}=\begin{cases}1&\,\text{if }Y^{*}_{1}>0,\\ 0&\,\text{otherwise},\end{cases}$

NTCIR12-MathWiki-8rate: 2

$Y_{2}=\begin{cases}1&\,\text{if }Y^{*}_{2}>0,\\ 0&\,\text{otherwise},\end{cases}$

NTCIR12-MathWiki-8rate: 4

$f(x)=\begin{cases}\exp(-1/x)&\,\text{if }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}$

NTCIR12-MathWiki-8rate: 4

$w=\begin{cases}w^{*}&\mbox{if }~{}w^{*}>\frac{1}{2},\\ \frac{1}{2}&\mbox{if }~{}w^{*}\leq\frac{1}{2}.\\ \end{cases}$

NTCIR12-MathWiki-8rate: 0

$A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},$

NTCIR12-MathWiki-8rate: 1

$y=\begin{cases}0&\,\text{if }y^{*}\leq\mu_{1},\\ 1&\,\text{if }\mu_{1}<y^{*}\leq\mu_{2},\\ 2&\,\text{if }\mu_{2}<y^{*}\leq\mu_{3},\\ \vdots\\ N&\,\text{if }\mu_{N}<y^{*}.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$y=\begin{cases}0~{}~{}\,\text{if}~{}~{}y^{*}\leq 0,\\ 1~{}~{}\,\text{if}~{}~{}0<y^{*}\leq\mu_{1},\\ 2~{}~{}\,\text{if}~{}~{}\mu_{1}<y^{*}\leq\mu_{2}\\ \vdots\\ N~{}~{}\,\text{if}~{}~{}\mu_{N-1}<y^{*}.\end{cases}$

NTCIR12-MathWiki-8rate: 1

$y=\begin{cases}1~{}~{}\,\text{if}~{}~{}y^{*}\leq\theta_{1},\\ 2~{}~{}\,\text{if}~{}~{}\theta_{1}<y^{*}\leq\theta_{2},\\ 3~{}~{}\,\text{if}~{}~{}\theta_{2}<y^{*}\leq\theta_{3}\\ \vdots\\ K~{}~{}\,\text{if}~{}~{}\theta_{K-1}<y^{*}.\end{cases}$

NTCIR12-MathWiki-8rate: 0

$A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},$

NTCIR12-MathWiki-8rate: 0

$\omega=\begin{cases}\sqrt{D}&\mbox{if }~{}D\equiv 2,3\;\;(\mathop{{\rm mod}}4)% \\ {{1+\sqrt{D}}\over 2}&\mbox{if }~{}D\equiv 1\;\;(\mathop{{\rm mod}}4)\end{cases}$

NTCIR12-MathWiki-8rate: 0

$p\cdot q=\textstyle\frac{1}{2}(p^{*}q+q^{*}p)=\textstyle\frac{1}{2}(pq^{*}+qp^% {*}).$

NTCIR12-MathWiki-8rate: 3

$\mathrm{rect}(t)=\Pi(t)=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}$

NTCIR12-MathWiki-8rate: 1

$\hat{F}(x)=\begin{cases}\Phi(\hat{w})+\phi(\hat{w})(\frac{1}{\hat{w}}-\frac{1}% {\hat{u}})&\,\text{for }x\neq\mu\\ \frac{1}{2}+\frac{K^{\prime\prime\prime}(0)}{6\sqrt{2\pi}K^{\prime\prime}(0)^{% 3/2}}&\,\text{for }x=\mu\end{cases}$

NTCIR12-MathWiki-8rate: 2

$\mathbf{a}_{i}=\begin{cases}1&\,\text{if }w_{i}>0,\\ 0&\,\text{if }w_{i}<0.\end{cases}$

NTCIR12-MathWiki-8rate: 0

${\mathit{l}\over\mathit{l}^{*}}={1\over 2}$

NTCIR12-MathWiki-8rate: 0

$\chi_{A}(x)=\begin{cases}1&\mbox{if }~{}x\in A,\\ 0&\mbox{if }~{}x\notin A.\\ \end{cases}$

NTCIR12-MathWiki-8rate: 1

$f_{\mathrm{img}}=\begin{cases}f+2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}% >f\mbox{ (high side injection)}\\ f-2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}<f\mbox{ (low side injection)}% \end{cases}$

NTCIR12-MathWiki-8rate: 2

$f:\mathbb{R}\to\mathbb{R};\qquad f(x)=\begin{cases}e^{-\frac{1}{x^{2}}}&x>0,\\ 0&x\leq 0.\end{cases}$

NTCIR12-MathWiki-8rate: 2

$y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>y_{L}\\ y_{L}&\textrm{if}\;y_{i}^{*}\leq y_{L}.\end{cases}$

NTCIR12-MathWiki-8rate: 4

$y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}$

NTCIR12-MathWiki-8rate: 0

$\mathcal{|}a|=\begin{cases}a,&\textrm{if }a^{(p-1)/2}\equiv 1\bmod{p},\\ -a,&\textrm{if }a^{(p-1)/2}\equiv-1\bmod{p}.\end{cases}$

NTCIR12-MathWiki-8rate: 2

$g=\begin{cases}-\sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T>\frac{1}{% 2}\\ \sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T<\frac{1}{2}\\ 0,&\mbox{if}~{}~{}T=0\\ \end{cases}$

NTCIR12-MathWiki-8rate: 2

$I_{|X|\geq K}=\begin{cases}1&\,\text{if }|X|\geq K,\\ 0&\,\text{if }|X|<K.\end{cases}$

NTCIR12-MathWiki-9rate: 2

${I_{1}\choose I_{2}}=\begin{pmatrix}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{pmatrix}{V_{1}\choose V_{2}}$

NTCIR12-MathWiki-9rate: 0

$\left[\mathbf{S}\right]=\begin{bmatrix}S_{11}&S_{12}\\ S_{12}&S_{11}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 2

$\begin{bmatrix}FR_{1}\\ FR_{2}\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}DP_{1}\\ DP_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\mathbf{S}(p)=\begin{bmatrix}s_{11}&s_{12}\\ s_{21}&s_{22}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\mathbf{P}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix}.$

NTCIR12-MathWiki-9rate: 0

$\begin{vmatrix}b_{1}&a_{12}\\ b_{2}&a_{22}\end{vmatrix}=\begin{vmatrix}a_{11}x_{1}&a_{12}\\ a_{21}x_{1}&a_{22}\end{vmatrix}=x_{1}\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}$

NTCIR12-MathWiki-9rate: 2

$\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}0&n\\ -n&0\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}V_{1}\\ 0\end{bmatrix}=\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\ -R_{2}&R_{2}+R_{3}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}=\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 3

$\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}z\\ v\end{bmatrix}=\mathbf{P}(s)\,\begin{bmatrix}w\\ u\end{bmatrix}=\begin{bmatrix}P_{11}(s)&P_{12}(s)\\ P_{21}(s)&P_{22}(s)\end{bmatrix}\,\begin{bmatrix}w\\ u\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$H=\begin{bmatrix}H_{11}&H_{12}\\ H_{12}^{\ast}&H_{22}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}v\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}i\\ u\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

${V_{1}\choose V_{2}}=\begin{pmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{pmatrix}{I_{1}\choose I_{2}}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}f_{1}\\ g_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}0\\ y_{0}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$T=\begin{bmatrix}T_{11}&T_{12}\\ 0&T_{22}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$A\Lambda=\begin{bmatrix}1&-3\\ 1&2\end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\end{bmatrix}.\,\!$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}l_{11}&0\\ l_{21}&l_{22}\end{bmatrix}\begin{bmatrix}u_{11}&u_{12}\\ 0&u_{22}\end{bmatrix}.$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}^{-1}$

NTCIR12-MathWiki-9rate: 0

$\mathbf{C}=\begin{pmatrix}\mathbf{C}_{11}&\mathbf{C}_{12}\\ \mathbf{C}_{21}&\mathbf{C}_{22}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\ \end{pmatrix}\begin{pmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\\ \end{pmatrix}=\mathbf{A}\mathbf{B}$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}V\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I\\ v\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}i\\ u\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}v\\ F\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}V_{1}\\ V_{0}\end{bmatrix}=\begin{bmatrix}z(j\omega)_{11}&z(j\omega)_{12}\\ z(j\omega)_{21}&z(j\omega)_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{0}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$M=\left(\begin{matrix}{{M}_{11}}&{{M}_{12}}\\ {{M}_{21}}&{{M}_{22}}\\ \end{matrix}\right)$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{1% 6}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}% \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 1

NTCIR12-MathWiki-9rate: 3

$\begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}K_{11}&K_{12}\\ K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}F_{1}\\ F_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{pmatrix}B\\ C\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}A\\ D\end{pmatrix}.$

NTCIR12-MathWiki-9rate: 1

$\begin{pmatrix}b_{1}\\ a_{1}\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}a_{2}\\ b_{2}\end{pmatrix}\,$

NTCIR12-MathWiki-9rate: 1

$\begin{pmatrix}b_{1}\\ b_{2}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}a_{1}\\ a_{2}\end{pmatrix}\,$

NTCIR12-MathWiki-9rate: 1

$\begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}=\begin{bmatrix}h_{1}&-h_{2}\\ h_{2}^{*}&h_{1}^{*}\end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2}^{*}\end{bmatrix}+\begin{bmatrix}n_{1}\\ n_{2}^{*}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\end{bmatrix}=\begin{bmatrix}\sigma_{11}&\sigma_{21}&\sigma_{31}\\ \sigma_{12}&\sigma_{22}&\sigma_{32}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{bmatrix}\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{13}\\ \sigma_{23}\end{bmatrix}\,dx_{3}\,.$

NTCIR12-MathWiki-9rate: 0

$G=\begin{pmatrix}G_{11}&G_{12}\\ G_{21}&G_{22}\end{pmatrix}$

NTCIR12-MathWiki-9rate: 0

NTCIR12-MathWiki-9rate: 4

$\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 2

$\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}$

NTCIR12-MathWiki-9rate: 0

$\begin{bmatrix}y_{1,t}\\ y_{2,t}\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\end{bmatrix}+\begin{bmatrix}A_{1,1}&A_{1,2}\\ A_{2,1}&A_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1}\\ y_{2,t-1}\end{bmatrix}+\begin{bmatrix}e_{1,t}\\ e_{2,t}\end{bmatrix},$